W K Burton and Other on N CAbrera

[ 299 ]
THE GROWTH OF CRYSTALS AND THE EQUILIBRIUM
STRUCTURE OF THEIR SURFACES
By W . K . B U R T O N * , N . C A BR E R A
F. C. F R A N K
and
H. H. Wills Physical
abort , University of Bristol
L
Communicatedby N. F.
Revised 25 July 1950 —Read 2 November 1950)
(i
,M
ot F.R.S.—Received 15 March 195
CONTENTS
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PAGE
A THEORY OF GROWTH OF REAL
CRYSTALS
P art I. M ovement
300
1. Introduction
2 . Mobility of adsorbed molecules on a
crystal surface
3. Concentration of kinks in a step
4. Rate of advance of a step
5. Parallel sequence of steps
6. The rate of advance of small closed
step-lines
P art IV. Structure
302
303
304
308
EQUILIBRIUM STRUCTURE OF
CRYSTAL SURFACES
Steps
and
17. Introduction
18. Co-operative phenomena in crystal
lattices
19. Two-level model of a crystal surface
20. Many-level model: Bethe’s approxi­
mation
20T. Two-level problem
20*2. Three-level problem
20*3. Many-level problem
308
310
APPENDICES
313
313
316
317
319
322
A ppendix A.
(Price 14J.)
335
336
340
342
343
344
345
I nfluence
of the mean
OF ADVANCE OF STEPS
345
A l. Single step
A 2. Parallel sequence of steps
345
347
A ppendix B. T he
mutual influence of
347
A PAIR OF GROWTH SPIRALS
324
A ppendix C. P roof of certain
324
325
formulae
IN THE STATISTICS OF KINKS
349
A ppendix D. W ulff’s theorem
351
A ppendix E. A n
METHOD
OF
outline of the matrix
TREATING
CO-OPERATIVE
354
PROBLEMS
327
329
E l. The one-dimensional, two-level
case
E2. The two-dimensional, two-level
case: rectangular lattice
330
355
356
358
R eferences
330
., Butterwick Research Laboratories, The Frythe,
* Seconded from Imperial Chemical Industries
Welwyn, Herts.
Vol. 24.3. A. 866.
334
DISTANCE X0 BETWEEN KINKS ON THE RATE
324
13. Introduction
14. Equilibrium structure of a step
14T. Equilibrium structure of a
straight step
14-2. Free energy of steps
15. The two-dimensional nucleus: activa­
tion energy for nucleation
15T. The shape of the equilibrium
nucleus
334
311
two -dimensional
nuclei
333
of a crystal surface
AS A CO-OPERATIVE PHENOMENON
310
7. Introduction
8 . The growth pyramid due to a single
dislocation
9. The growth pyramids due to groups of
dislocations
9T. Topological considerations
9*2. General case
10. Rate of growth from the vapour
11. Comparison with experiment
12. Growth from solution
332
300
of growth of a crystal
surface
Part III.
300
of steps on a crystal
surface
Part II. R ates
PAGE
Part III (
cont). .
15*2. Activation energy for twodimensional nucleation
16. Steps produced by dislocations
40
[Published 12 June 1951
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300
W. K. B U R T O N A N D O T H E R S O N
Parts I and II deal with the theory of crystal growth, parts III and IV with the form (on the atomic
scale) of a crystal surface in equilibrium with the vapour. In part I we calculate the rate of advance of
monomolecular steps (i.e. the edges of incomplete monomolecular layers of the crystal) as a function
of supersaturation in the vapour and the mean concentration of kinks in the steps. We show that in
most cases of growth from the vapour the rate of advance of steps will be independent of their
crystallographic orientation, so that a growing closed step will be circular. We also find the rate
of advance for parallel sequences of steps, and the dependence of rate of advance upon the curvature
of the step.
In part II we find the resulting rate of growth and the steepness of the growth cones or growth
pyramids when the persistence of steps is due to the presence of dislocations. The cases in which
several or many dislocations are involved are analysed in some detail; it is shown that they will
commonly differ little from the case of a single dislocation. The rate of growth of a surface containing
dislocations is shown to be proportional to the square of the supersaturation for low values and to
the first power for high values of the latter. Volmer & Schultze’s ( 1931) observations on the rate of
growth of iodine crystals from the vapour can be explained in this way. The application of the same
ideas to growth of crystals from solution is briefly discussed.
Part III deals with the equilibrium structure of steps, especially the statistics of kinks in steps, as
dependent on temperature, binding energy parameters, and crystallographic orientation. The
shape and size of a two-dimensional nucleus (i.e. an ‘island* of new monolayer of crystal on a
completed layer) in unstable equilibrium with a given supersaturation at a given temperature is
obtained, whence a corrected activation energy for two-dimensional nucleation is evaluated. At
moderately low supersaturations this is so large that a
crystal would have no observable growth
rate. For a crystal face containing two screw dislocations of opposite sense, joined by a step, the acti­
vation energy is still very large when their distance apart is less than the diameter of the corre­
sponding critical nucleus; but for any greater separation it is zero.
Part IV treats as a ‘co-operative phenomenon’ the temperature dependence of the structure
of the surface of a perfect crystal, free from steps at absolute zero. It is shown that such a surface
remains practically flat (save for single adsorbed molecules and vacant surface sites) until a transition
temperature is reached, at which the roughness of the surface increases very rapidly (‘
melting’). Assuming that the molecules in the surface are all in one or other of two levels, the results
of Onsager ( 1944) for two-dimensional ferromagnets can be applied with little change. The transition
temperature is of the order of, or higher than, the melting-point for crystal faces with nearest
neighbour interactions in both directions (e.g. ( 100) faces of simple cubic or ( 111) or ( 100) faces of
face-centred cubic crystals). When the interactions are of second nearest neighbour type in one
direction (e.g. ( 110) faces of s.c. or f.c.c. crystals), the transition temperature is lower and corre­
sponds to a surface melting of second nearest neighbour bonds. The error introduced by the
assumed restriction to two available levels is investigated by a generalization of Bethe’s method
( 1935) to larger numbers of levels. This method gives an anomalous result for the two-level problem.
The calculated transition temperature decreases substantially on going from two to three levels,
but remains practically the same for larger numbers.
Note on authorship. Although at all times a constant interchange o f ideas took place between
all three authors, the principal contributions o f one o f us (F.C.F.) are to part II o f this paper.
Part IV , and all the calculations in parts I and III, are due exclusively to W .K.B. and N.G.
A THEORY OF GROWTH OF REAL CRYSTALS
P art I. M ovement of steps on a crystal surface
1. Introduction
The theory o f growth o f perfect crystals has been developed extensively during the past thirty
years, especially by the work o f Volmer (1939), Stranski (1928, 1934), and Becker & Doring
(1935). The essential ideas were put forward earlier by Gibbs (1878).
T H E G R O W T H O F C R Y STA LS
301
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According to this theory, w hen all surfaces o f high index (stepped surfaces) have dis­
appeared, the crystal will continue to grow by two-dim ensional nucleation o f new molecular
layers on the surfaces o f low index (saturated surfaces). As in all nucleation processes, the
probability for the formation o f these tw o-dim ensional nuclei is a very sensitive function o f
the supersaturation. This probability is quite negligible below a certain critical supersatura­
tion and increases very rapidly above it. Assuming reasonable values for the edge energy o f
the two-dim ensional nuclei, one recognizes that this critical supersaturation should be o f the
order o f 50 %. A t the supersaturations at w hich real crystals grow (1 % and even lower) the
probability o f formation o f nuclei should be, according to this theory, absolutely negligible
(Burton, Cabrera & Frank 1949; Burton & Cabrera 1949, cf. also part III).
Figure 1. The presence of a kink in a step on a crystal surface.
R ecently, Frenkel (1945) pointed out that the structure o f a perfect crystal surface above
the absolute zero o f temperature w ould have a certain roughness produced by thermal
fluctuations. H e discussed the structure o f a m onom olecular
and proved that it will
contain a high concentration o f kinks, illustrated in figure 1, and introduced before by
Kossel (1927) and Stranski (1928). Burton & Cabrera (1949, cf. also part III) have shown
that the concentration o f kinks is even larger than was supposed by Frenkel; this result is very
important from the point o f view o f the rate o f advance o f the steps, which will be developed
in part I o f this paper. O n the other hand, Frenkel generalized this idea to the formation o f
steps in a perfect crystal surface, but Burton & Cabrera (1949, cf. also part IV ) have shown
that steps will not be created by therm odynam ical fluctuations in a low-index crystal surface,
unless, perhaps, close to the m elting-point; therefore the steps required for growth can only
be produced, on a perfect crystal surface, under a highly supersaturated environment.
W e conclude that the growth o f crystals under low supersaturations can only be explained
by recognizing that the crystals which grow are not perfect, and that their imperfections (in
particular, dislocations terminating in the surface with a screw component) will provide the
steps required for growth, making two-dimensional nucleation unnecessary. This idea,
introduced by Frank (Burton et al. 1949; Frank 1949) will be developed in this paper, and
we shall see that it explains most o f the features o f crystal growth at low supersaturations.
This theory o f growth o f real crystals assumes the existence o f dislocations in them, but
does not depend critically on their concentration. T he study o f crystal growth should perhaps
also explain the formation o f dislocations which, as in the case o f steps in a crystal surface,
cannot be due to thermodynamical fluctuations. Several mechanisms can be visualized for
4 0 -2
302
W. K. B U R T O N A N D O T H E R S O N
the formation of new dislocations during growth (Frank 1949), but no detailed theory has
yet been formulated.
Mobility of adsorbed molecules on a crystal surface
2.
We know that in general a crystal surface in contact with its vapour will contain a certain
concentration ns per cm .2 o f adsorbed, essentially m obile molecules. Under equilibrium
conditions, the concentration ns0 o f adsorbed molecules will be given by a formula o f the
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ns0 = n0exp (-W JkT ), (1)
where Ws is the energy of evaporation from the kinks on to the surface; nQcontains entropy
factors, but in simple cases will be o f the order o f the number per cm .2 o f molecular positions
on the surface.
The process o f growth o f a crystal surface with steps will be the result o f three separate
processes: (i) exchange o f molecules between adsorbed layer and vapour, (ii) diffusion o f
adsorbed molecules towards the steps and exchange with them, and (iii) perhaps also
diffusion o f adsorbed molecules in the edge o f the steps toward the kinks and exchange with
them.
In order to discuss the role o f the diffusion on the surface we must introduce the mean
displacement xs of adsorbed molecules. This can be defined in quite general terms by Einstein’s
f° rmula:
'
*? = 4 r s,
(2)
where Ds is the diffusion coefficient and ts the m ean life o f an adsorbed m olecule before being
evaporated again into the vapour. For simple molecules we can write
A = a V exp ( — UJk T ),
(3)
llTs = vexp(-W;/kT),(4)
and
where Us is the activation energy between two neighbouring equilibrium positions on the
surface, distant a from each other, and
W'sthe evaporation en
vapour. The frequency factors v' and v would both be o f the order o f the atomic frequency
of vibration [v ~ 1013sec._1) in the case o f monatom ic substances, but they will be different
in the case o f more complicated molecules. Using (3) and (4), (2) becomes
— <zexp {(
W] —Us)/ 2kT},
assuming v' ~ v. T he condition for the diffusion on the surface to play an important role is
that
xs>
,a and therefore from (5), W '> Us. This is probably always the case; then xs can be
much larger than a and increases rapidly as the temperature decreases.
In order to have an idea o f the values that we can expect for xs, let us consider, for instance,
a (1 ,1 ,1 ) close-packed surface o f a face-centred cubic crystal. By simple considerations
regarding the interaction 0 between nearest neighbours only, W' — 30, i.e. half the total
evaporation energy W = Ws+W 's; while Us ~ p = \W . However, fuller calculations carried
out by M ackenzie (1950) using Lennard-Jones forces show that Us is considerably smaller,
about
fo W
.H ence, in this case, (5) becomes
(3(f>l2kT) ~4 x 102a
;v5~ a e x p
for the typical value
4.
(6)
T H E G R O W T H O F C R Y STA LS
303
It is interesting to notice that
xswill be a function o f the crystal face
and Us being different for different faces. In general, xs will be smallest for the closest packed
surface, because W's increases more rapidly than
For instance, for a (1 ,0 ,0 ) surface in
a face-centred cubic (f.c.c.) crystal assuming nearest neighbour forces,
= 40 and Us is
probably still very small. T hen
xs~
aexp
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3. Concentration of kinks in a step
Frenkel (1945) and Burton & Cabrera (1949, cf. also part III) have shown that these steps
must always contain a large concentration o f kinks. In the case o f short-range intermolecular
forces, we can briefly summarize in the following manner the results o f the theory which have
an important bearing on growth.
Let a close-packed crystallographic direction be taken as the #-axis, and consider a step
w hich follows this axis in the m ean, so that the surface is one m olecule higher in the region
y<£.0 than in the region y^>0 . Follow ing the step along the direction o f increasing x, points
where y increases or decreases by a unit spacing a are called positive or negative kinks
respectively. For this orientation, the step contains equal numbers o f positive and negative
kinks, and their total number is less than for any other orientation. Let 2 and q be the
probabilities for having a kink or no kink, respectively, at a given site in the step. T hen we
must have
n/q= exp
where w is the energy necessary to form a kink. H ence the m ean distance #0 =
kinks is
x0 = |a{exp (w/k T) + 2} ~ 1 exp (wjk T ) ,
between
(7)
where a is the intermolecular distance in the direction o f the step.
As the inclination 6 o f the step relative to a close-packed direction increases, the number
o f kinks increases. Let n+ and n_ be the probabilities for having a positive or negative kink
respectively.
As an approxim ation, w e neglect the difference between q and unity. Then
2 n = n++n_,
6 = n+ —n_,
where n is the probability for having a kink o f any kind and 6 is assumed to be small. For any
inclination, and from therm odynam ical considerations,
n+n_
—exp ( —2
Thus the m ean distance xQ(0 ) between kinks will now be
x0{6 ) = x0{l —%(xja)2 d2}}
(8)
assuming
6 < a/x 0and *0 is given by (7).
N o detailed calculation o f w has yet been made, but simple considerations suggest that w
must be a small fraction o f the evaporation energy. For instance, in a close-packed step in
a (1 ,1 ,1 ) face o f a f.c.c. crystal, w ith nearest neighbour interactions 0, it is easy to see that
w is equal to a quarter o f the energy necessary to move one molecule from a position in the
( —w/kT
304
W. K. B U R T O N A N D O T H E R S O N
straight step to an adsorption position against the straight step, equal to 2^; hence
or T2^) and the mean distance between kinks is, from (7),
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x0= \a exp (<j)\2k T) ~ 4a,
(9)
for the typical value (j)JkT~ 4.
The concentration o f kinks in the steps will remain practically unchanged even if the
vapour is supersaturated. Hence the problem o f the rate o f advance o f a step is reduced to
a classical diffusion problem on the surface. T he important ratio in this calculation is xs/x0.
From (5) and (7) this is approximately 2 exp
W ith the estimates o f
W', Us and w which we have m ade above on the basis o f the simple model o f a close-packed
homopolar crystal, this is about 102. Thus it appears that we m ay generally assume xs #0,
in which case we can perform the diffusion calculation regarding the step as a continuousline sink. It is an interesting corollary o f this case that the rate o f advance o f a step is then
independent o f its crystallographic orientation. However, the estimate is uncertain for
various reasons, such as the neglect o f entropy factors, and we shall also examine the cases
in which #0 is comparable with or larger than xs.
Rate of advance of a step
4.
T he supersaturation a in the vapour is defined as
a= a —1,
where p is the actual vapour pressure, p 0 the saturation value, and a will be called the
saturation ratio. W e assume a to be constant above the surface. There will also be a super­
saturation as o f adsorbed molecules on the surface (in general, dependent on position)
defined by
,
,
h*a = wA o >
(11)
where ns and ns0 are the actual and equilibrium concentration o f adsorbed molecules
respectively.
The equations governing the diffusion o f adsorbed molecules towards the step are easily
written down. The current on the surface will be
js
= -Q g r& d n s ~ Q n s0 gr<idf 9f =
(12)
where Ds is the diffusion coefficient o f adsorbed molecules. There will also be a current j v
going from the vapour to the surface, which is easily seen to be
jv =
(*.-«»)
where ts is the mean life o f an adsorbed molecule on the surface, defined in § 2.
Let us make the assumption (to be justified a posteriori) that the movement o f the step can
be neglected in the diffusion problem, so that the adsorbed molecules have a steady distribu­
tion on either side o f the step which is practically the same as though the step were absorbing
molecules without moving. U nder these conditions ijr must satisfy the continuity equation
divj, = j vi
or using (6) and (7) and assuming Ds independent o f direction in the surface
x2 y 2^. _ ^
where xs is the mean displacement o f adsorbed molecules, defined in § 2.
(14)
T H E G R O W T H O F C R Y STA LS
305
N ow if we compare the typical values for xs and *0 estimated in (6) and (9) respectively,
we see that in most cases, certainly for m onatom ic substances, xs |> #0. U nder these conditions,
each molecule deposited from the vapour on the surface near the step will have a high
probability to reach a kink in the step before being evaporated again into the vapour.
Therefore the concentration o f adsorbed molecules near the step will be controlled by
evaporation from and condensation into the kinks. T hen, provided this exchange is very
rapid, the concentration near the step should be m aintained equal to the equilibrium value,
independently o f the supersaturation existing in the vapour.
as= 0 near the step, and crs = <r far from it, equatio
Thus, if we assume that
simple solution
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^ = < rex p (T y /* J ,
(15)
where
y is the distance to the step, the minus sign being used for y > 0 and the plus sign for
y < 0 . N ow the current
jgoing into the step per cm. per sec. will be obtained from (12) usin
(15) and putting
y = 0. T he velocity o f the step is then » where l/n 0 is the area per
molecular position; therefore
t>oo
= 2axsv exp ( —W/kT(16)
where expressions (1) and (4) for ns0 and ts have been used, and
is the total
evaporation energy. T he factor 2 comes about because o f the contribution from
0 and
y < 0. T he advance o f the step is therefore ow ing to the m olecules condensing from the
vapour on a strip o f w idth xs at both sides o f the step. This expression represents the m axim um
velocity o f a step in a given direction, and if Ds and therefore xs were independent o f direction,
then the velocity o f the step w ould be independent o f its orientation.
W e can now justify our neglect o f the m otion o f the step w hen treating the diffusion
problem. This is permissible if the characteristic distance {DJv^) is great compared with the
characteristic distance xs. N ow , from (3), (5) and (16),
vo0xJDs
=
2 a exp ( —WJkT) ~ 2<rexp ( —12) <^1.
W e believe that (16) is correct, at least in the case o f m onatom ic substances. In the case o f
molecular substances we must introduce tw o possible complications: (i) as W yllie (1949)
has pointed out the exchange betw een the kinks and the adsorbed layer m ight not be rapid
enough to m aintain <rs
—0 near the kinks; (ii) the condition xs&x 0 is perhaps not s
It is easy to see that (i) introduces a supplem entary factor
1 in formula (16) given by
+ xsTlaTs)~(17)
where r is the time o f relaxation necessary to re-establish equilibrium near the step. The
supersaturation near the step will then be <r4(0) = (1 —fi) a; /? will be smaller than 1, for
instance, when the rotational entropy o f the adsorbed molecules is much larger than that o f
the molecules in the solid. I f condition xs^>x0 is nevertheless satisfied, then vmshould still be
independent o f the orientation o f the step.
O n the other hand, if
is not satisfied, the supersaturation near the step will not be
constant, and will be a function o f #0. T he other extreme case, when
is easy to consider.
It is then necessary to discuss the influence o f the diffusion o f adsorbed molecules in the edge
o f the step. I f the contribution o f the current via the edge is important, then this will help
306
W. K. B U R T O N A N D O T H E R S O N
to keep the supersaturation constant near the step, even if #0|> T he most unfavourable case
would be when the current via the edge can be neglected altogether. Then, assuming Ds to
be independent o f direction, the required solution o f (14), around an isolated kink on the
step, is
\jr{r) = /for KqW x,)
K 0(a/xs)
where K0 is the Bessel function o f second kind with imaginary argument and order zero, and
we assume that the supersaturation o f adsorbed molecules is maintained equal to (1 —/?) cr
at a distance
r — afrom the kink. The current going into every step is then easily calculat
and the velocity o f advance o f the step is given by
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va>= {2<r/fa5yexp { —W /kT )} 7rxs/x0ln
where we assume always xs> a and we use the approximate formulae
KM xs) = ln (2
= -x ja ,
y = T78 being Euler’s constant. W e m ay verify once again that, at least for sufficiently small
supersaturations, we may neglect the motion o f the kink in treating the corresponding
diffusion problem. The criterion this time is (v^in^xs/Ds) <^.1, where vkink = v^xja. N ow by
use o f (3) and (5)
* M = [27T/?«r exp { - ( W + J Us - f W's) jh 7 }] /{In (2 //) + ( W. - U s)lk T}.
T he exponential factor here remains considerably less than unity for any reasonable estimate
o f the energies, especially as the condition x0> x 5 is only likely to arise when Us is unusually
large or
W'su nusually small, though Us can scarcely exceed W' so as to make the denominator
small.
W e see that apart from the factor /?, the m axim um velocity (16) is multiplied by another
factor
c0< 1, given by
^ = irxjxQln (2
(18)
In the general case, the velocity of advance o f the step can also be represented by the general
formula
= 2 axsvexp ( —W/kT) fic0)
(19)
where c0 is between 1 and the value given by equation (18).
The calculation o f c0 in the general case is a difficult problem. The easiest w ay to solve it is
to assume that there is a diffusion in the edge o f the step and that it is important enough for
the current going directly from the surface to the kinks to be neglected. In appendix A we
treat along these lines the problem o f a single step with equally spaced kinks.
The relative importance of the current diffusing in the edge o f the step and that diffusing
on the surface will be represented by the non-dimensional factor
D eneolD s nsOa >
where De and ne0 are the diffusion coefficient and the equilibrium concentration respectively
o f adsorbed molecules in the edge. This factor is equal to (
) 2, where xe is the mean dis­
placement o f adsorbed molecules in the edge. Actually x\ —Derei and by definition
7e= ne0alDsns0 = (1/.) exp {(Wt -f Us)jk T ];
therefore
x* = Dene0a/Dsns0 ~ ^ exp {(W'e + Us- Ue) \k T},
(2 0 )
307
where W' is the energy necessary to take an adsorbed m olecule from the edge to the surface
and Ue is the activation energy for diffusion in the edge.
It is clear that
Ds> D e; on the other hand, ns0a < n eQ. I f xe> a , the current going into
kinks goes essentially via the edge and the point o f view adopted above is justified. If, on
the contrary,
xe<
a, the im portant contribution is that due to direct condensation from the
surface into the kinks. T he case xe~ a can also be interpreted as if the edge did not exist at all,
since this w ould correspond to W'e = 0, Ue — Us, w hich implies also xe = a. H ence the
m ethod o f calculation suggested above should give for
the same result as if the influence
o f diffusion in the edge were neglected altogether.
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e
T H E G R O W T H O F C R Y STA LS
2. The factor c0 as a function of
xja for the values of xja indicated on the curv
It is difficult in general to estimate xe. In the particular case o f a close-packed step in a
(1 ,1 ,1 ) face o f a f.c.c. crystal, with nearest neighbour interactions
we can estimate
Wg = 2
(j),Us~ 0 , Ue~ 2<j>, hence
xe~ a. Assuming this to be the cas
given in appendix A reduces to
l/r 0 = l + 2 H n { ( 4 6 x » / ( l + (l + 62)*)},
=
(21)
for several values o f xja. It is intei*esting
which is represented in figure 2 as a function o f
to notice that if #0 increases indefinitely in (21), then
c0 =
7txs/ x 0 In
(4
which practically coincides with (18) as we should expect. W e notice that c0 differs appreciably
from unity for x0^ x s.
Vol. 243. A.
41
W. K. B U R T O N A N D O T H E R S O N
308
In conclusion to this paragraph, and from the estimates o f and x0 made in §§ 2 and 3,
we deduce that in most cases o f growth from the vapour the rate of advance of the step must be
practically independent of its
ta. In some cases, perhaps, the factor c0 in (19) c
orien
smaller than 1 for the close-packed slowest steps, containing a m inimum number o f kinks.
As the orientation o f the step deviates from that o f closest packing, c0 will become rapidly
equal to 1.
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5.
Parallel sequence of steps
Another interesting problem is that o f the m ovem ent o f a parallel sequence o f steps
separated by equal distances y0.
I f we assume that the distance x0 between kinks in every step satisfies the condition
x04 xs) and that near every step crs = 0, the solution o f equation (14) is easily seen to be
, Cosh (yf e )
cosh (;
(22)
between two steps, where y is the distance from the m id-point between two steps. The
current going into every step is again calculated from (12), using (22) and putting y = Jy0;
the velocity o f every step is then
vm =
2axsv exp ( — W/k T) tanh (y0/ 2xs) ,
which reduces to (16) if
y0->
oo.
In the general case where (i) the interchange with the kinks is not rapid enough to
maintain <rs
= 0 near the step and (ii) the condition x0<^xs is not satisfied, we obtain again
a general formula o f the type
Voo = 2 axsvexp ( — W/k T) tanh (y0/ 2xs) /fc0,
(24)
where c0< 1 is a function o f x0 and y0. T he calculation o f c0 is now rather complicated.
In the particular case when
xe~
ain the edge o f the steps, one can give fo
mated expression
1/c0 =
b=
l + 2 banh
t
(yjx s) [ l n { ( 4 £ * » / ( l + (1 + 6 2)*)}+ (2xs/y0) tan"16],
x0/2iTXS,
For y0-> oo this expression reduces o f course to (21). As
decreases, c0 in general will be
nearer 1 than (21) is, as we should expect, but this does not mean that for a sufficiently small
value of
y 0we shall get r0 — 1> because o f the second term in the parentheses in (25).
6. The rate of advance of small closed step-lines
W e know that, given a certain supersaturation in the vapour, there will be a certain twodimensional nucleus which is in unstable equilibrium with it. The shape and size o f this
critical nucleus are perfectly defined; its shape has been studied in detail in one special case,
Burton et al. 1949; cf. also part III. It is interesting to notice at this point that the number o f
kinks per unit length in every position o f the edge o f the critical nucleus is the same as that
in an infinite step having the same orientation.
If the nucleus is larger than the critical one it will grow. Then its shape will be determined
essentially by the differences in velocity for the different orientations and not by thermo-
T H E G R O W T H O F C R Y STA LS
309
dynam ical considerations. In particular, if the velocity is the same for all orientations, it
will becom e circular. T he study o f the shape o f a growing nucleus o f large dimensions will be
considered in detail by one o f us (F. C. F.) in a later paper; here we are concerned with
the absolute value o f the velocity w hen the dimensions are not very different from those o f
the critical nucleus.
Let us consider first the case o f independence o f velocity on orientation, when the nucleus
is circular. W e shall consider afterwards w hat changes can be expected when the velocity
depends on the orientation.
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T he m ean evaporation from the nucleus to the surface will be a function o f its dimensions.
I f its radius is
p,and we define an edge energy o f the nucleus equal to y per m olecule, the
m ean energy o f evaporation will be
m p ) = W -{y a lp ),
where a is as usual the interm olecular distance.
Let a be the supersaturation in the vapour; the nucleus will grow, but at every m om ent a
steady distribution o f adsorbed molecules will be formed provided the nucleus is not too
small. T he influence o f the m ovem ent o f the boundary on the diffusion problem can be
neglected for the low supersaturations considered. T he steady supersaturation in the surface
will satisfy the continuity equation (14) w hich now becomes
x2 W2i/r(r) — ^(r),
i/r(r) = cr—(rs(r).
(26)
T he solution o f this equation is
(27)
where / 0 and K 0 are the Bessel functions o f first and second kind with imaginary argument.
i/r(p) is the value o f ijr near the edge o f the nucleus. I f the interchange o f molecules between
the nucleus and the surface is rapid enough, the supersaturation
) near the edge o f the
nucleus will be m aintained equal to that which would be in equilibrium with it. This super­
saturation ffs(p) is given by
therefore
tip ) =
(28)
= a —[exp {yajpkT) —!].
The current going into the nucleus is now
j =
TpDsns0(difrldr)r:=p=
2 iK0(p/xs),
where the formulae
/<!(*) = /.(z ),
K'0(z) = —Kt(z),
K,{z) + 7 ,(2 ) K0(z) = 1/2
have been used. The radial velocity, v(p) = j( p ) / 2npn0, is therefore
v(p )
= x^v twp ( —WjkT) t{p)lpIo(filxs) U
pM
,
(29)
W. K. B U R T O N A N D O T H E R S O N
310
where we have used the expressions for xs, ts and ns0, given by (2), (4) and (1). Current and
velocity change sign when
<rs(pc)— <7, which defines the critical nucleus to
given by
pc — yajkT
, = l + <7.
cL
ln
(30)
When p > x s we can use the approximation
l*s)Ko(Plxs) =
h(P
so that (29) becomes
v(p) = va f t —ocpc,f>)l a,
a = l + (7,
(31)
or if a is small,
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v (p )= v „ (l-p jp ),
(32)
where
vnis the maxim um velocity (16) o f a straight step o f any orientation, calculated in § 4.
This formula will be valid down to p = pc, provided the supersaturation is low enough for pc
to be larger than xs, and ln a ~ c r . As <7 increases and pc decreases, v(p) will be a more
complicated function for p ~ pc.
It is also interesting to consider the rate o f advance o f a sequence o f concentric circles,
distant y 0 from each other. T he diffusion problem can also be solved in a similar way. The
radial velocity o f the circle o f radius p turns out to be
v(p) = 2(7*,Vexp ( - W /k T ) tanh (yQfexs) (1 - p j p ) ,
(33)
w h e n y0<^p and In a ~
a.We see that v(p) is again o f the general form (32), i
now the velocity (23) corresponding to a sequence o f parallel steps. W e see therefore that
the only change in the rate o f advance o f a large closed step o f radius p, with respect to that
o f an infinite step is the factor (1 —
pj).This is just what we should expect
supersaturation near the edge o f the nucleus is not zero, as in the case o f the straight step, but
OPciPThe same general formula (32) will apply also when (i) the interchange o f molecules
between nucleus and surface is not rapid, and (ii) the velocity depends on the orientation o f
the edge. T hen the nucleus will not be a circle, but the close-packed slowest orientations o f
the edge will move according to (32) where vn is now given by the general formula (24);
p will be the normal distance from the edge to the centre o f the nucleus and pc that corre­
sponding to the critical nucleus.
P art II. R ates of growth of a crystal surface
7. Introduction
In part I we have studied the movement o f steps on a perfect crystal surface, without
considering their origin. This information is sufficient to calculate the rates o f growth o f
stepped surfaces (those o f high index) where the steps exist because o f the geometry o f the
surface; nevertheless, it is clear that these steps will disappear in any finite crystal, after
a finite amount o f growth which completes the body bounded by close-packed, unstepped
surfaces circumscribed to the initial crystal.
Frank (1949) has shown that further growth o f these surfaces must be attributed to the
presence of steps associated with crystal defects, in particular dislocations having a component
o f displacement vector normal to the crystal face at which they emerge, ‘screw dislocations’
T H E G R O W T H O F C R Y STA LS
311
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for short, though they are not necessarily ‘pure screw ’ dislocations. H e has also shown that
w hen the crystal is growing under a supersaturated environm ent, the step due to a dislocation
winds itself in a spiral in such a w ay that a single screw dislocation sends out successive turns
o f steps (figure 3). I f the step is due to a right- and left-handed pair o f dislocations, they w ill
send out closed loops (figure 4) provided their distance apart is greater than the diameter
o f a critical nucleus. In both cases the dislocations will form pyramids and the concentration
o f step lines thereon will be large and practically independent o f the number o f disl ocations.
This provides an interpretation o f the pyramids o f vicinal faces long recognized as a normal
Figure 3. Growth pyramid due to a single screw
dislocation.
Figure 4. Growth pyramid due to a pair of
dislocations.
feature o f slow crystal growth (Miers 1903, 1904). In this part we shall study in more detail
the mathematics o f these pyramids (§§ 8, 9); once the rate at w hich these pyramids grow is
known we shall apply the results to growth from the vapour using the formulae for rate o f
advance o f steps deduced in I (§§10, 11). T he application o f the same ideas to growth from
solution will also be considered briefly in § 12. T he resulting topography o f the crystal
surface will be discussed by one o f us in a later paper.
8.
The growth pyramid due to a single dislocation
Let us first consider the spiral due to a single dislocation ending on an otherwise perfect
crystal surface. W e may suppose that, as new layers are added, the direction o f the dislocation
remains perpendicular to the surface, since this w ill usually m inimize the elastic energy. I f
the rate o f advance o f a step is independent o f its orientation (probably the case during
growth from the vapour, see I) the growing spiral will form a low cone, but it will tend to
form a pyramid when the rate o f advance depends on the orientation (as is illustrated in
figures 3 and 4). W e consider first the case o f a growing cone.
312
W. K. B U R T O N A N D O T H E R S O N
Following any increase in supersaturation, the step due to the dislocation will rapidly
wind itself up into a spiral centred on the dislocation, until the curvature at the centre reaches
the critical value
l/pc,at which curvature the rate o f advance falls to zero; the wh
will then rotate steadily with stationary shape.
We know (I, formula (32)) that the normal rate o f advance o f a portion o f spiral with
radius o f curvature p is given by
v(p) = v J l _ pJp)>
(34)
provided the supersaturation is not too high. Now let
represent the rotating spiral, in
(rotating) polar co-ordinates (r, 0). T he radius o f curvature at a point r will be
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p= (l+ rZ d'tyipd'+ rW s+ rd" ),
O' and
0 "being the derivatives o f
normal velocity at the point r is
, j = ^
0{r). If the angular velo
(36)
W e must now find
0 {r)and (o from these three equations.
A good approximation is obtained by taking an Archimedean spiral
r = 2 pc0 >O,
(37)
which has the proper central curvature, (o is then given by
a = »„/2
(38)
This approximation does not satisfy (34), especially for small r, but nevertheless gives a good
approximation to (o.
A better approximation can be obtained in the following w a y : one obtains the solutions
for small r (neglecting r2) and for large r (neglecting 1/r2) :
0 ' — l l 2 pc—(orlSv00pc,
r-> 0 :
(39)
6 '=
r-> oo:
Then, choosing a general expression o f the form
0' = a + 6 /( l+ r r ) ,
(41)
one determines,
a,b, c and co in such a way that (41) reduces to (39) and (40) for the
ranges o f
r.We obtain in this way
6 = 2(1 + 3*) [r/ ^ +ln t1 +'/*A).], .
co= 3*000/2^(1 + 3*) = O-630oo/2
(42)
\
This solution satisfies (34) within a few units per cent o f „ for all values o f p. The interesting
point is that the value o f (o obtained differs from (38) only by a factor close to 1, showing
that (o is insensitive to the actual law o f dependence o f on in the range in which p ~ pc.
Even the crudest approximation to (34),
P = PC
: *>= 0;
p > p c: v = vaj,
(Frank 1949) gives an angular velocity only twice as large as (38), i.e. just over three times
the best approximation (42).
T H E G R O W T H O F C R Y STA LS
313
In the rest o f this part, the number o f turns o f the spiral per second, <y/27r, will be called
the activity. T he actual rate o f vertical growth R o f the pyramid and therefore o f the crystal
will clearly be
R=
mfeir =
(43)
where
a=
n0Q is the height o f a step (n0 the number o f molecular positions per cm .2 in the
surface, Q the volum e o f a m olecule), and w e have used the approximation (38) for w.
T he distance y 0 betw een successive turns o f the spiral for large r is given by
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y 0 = 271/ 6 ' = lnpc.
(44)
These formulae will also be approxim ately valid w hen there is an influence o f the crystallo­
graphic orientation on the rate o f advance o f the steps. w is then the value corresponding to
the slowest advancing orientation, and pc h alf the dim ension o f the critical two-dimensional
nucleus.
9.
The growth pyramids due to groups of dislocations
9T . Topological considerations
W e now consider the interactions betw een the growth spirals centred on different dis­
locations. W e have already considered the case o f a pair o f opposite sign, and seen that if
they are closer together than a critical distance (2 in the simple case) no growth occurs,
w hile if they are further apart than this they send out successive closed loops o f steps. It is
obvious that if there are two such pairs these loops unite on m eeting, and the number o f steps
passing any distant point in a given tim e is the same as if only one pair existed. T he whole
area m ay be formally divided into two areas by a locus o f intersections o f the two families,
and one area m ay be considered to receive steps from the one centre, the other from the other
centre.
H ence two similar pairs o f dislocations o f opposite sign, separated by a distance large
compared with the separation in the pairs, have the same activity as one pair alone. Unless
the separation betw een pairs is a visible distance, there will be no macroscopic distinction
from the case o f one pair.
I f the two families o f loops are circles growing at the same rate the locus o f intersections
is a hyperbola in the general case, and in the symmetrical case a straight line bisecting the
line o f centres. Consideration o f the locus o f intersections, though trivial in the present case,
is useful for the treatment o f more com plex cases later. T he conclusion remains valid if the
loops are not circles, but, on account o f dependence o f growth rate on crystallographic
orientation, are deformed into polygons. T he same point applies in cases treated later.
W e chose to start w ith the case o f two opposed pairs as the simplest, since it can be topo­
logically analyzed in terms o f the locus o f intersections o f growing circles. W hen we consider
two simple
ldio
sn
cat, instead o f pairs, we have to consider the locus o f intersections o f spirals.
I f they are of opposite sign, a locus o f intersections still divides the area into two parts which
may be said to be fed with steps from each centre respectively. O f the possible loci o f inter­
sections, depending on the relative phase o f rotation o f the two spirals, the most important one
is that which is symmetrical— the bisector o f the line o f centres. For in this case there is
a possibility for an influence to be transmitted along each step from the point where turns o f
the two spirals meet, in to the respective centres, and there modify the rate o f rotation. I f
314
W. K. B U R T O N A N D O T H E R S O N
they meet nearer to the centre o f one than the other, there is a tendency for the rate o f
rotation of the former to be increased. So, in time, the two becom e synchronized in phase
and the locus o f intersections is the symmetrical one. As shown in appendix B, the increase in
rate o f advance o f the steps which is transmitted along the spirals from their m eeting point
to the neighbourhood o f their dislocation centres produces a small increase in the rate o f
rotation o f the spirals amounting probably at most to a few units per cent when the distance
between dislocations is o f the order o f 3
p ca nd decreasing exp
Then the activity o f the pair is indistinguishable from the activity o f o n e ; at the same time
there is no important topological difference from the case o f growing circles.
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\
\
\
\
Figure 5 . A pair of dislocations of like sign, separated by a distance l = AB > 2npc.
A
pairof dislocations of like sign gives a more com plex situation. I f they are far , a locus
o f intersections still divides the area into two parts, which m ay be said to be fed with steps
from each centre respectively. As before, there will be a tendency for the symmetrical case
to establish itself. T he locus o f intersections is then no longer a straight line, but an S-shaped
curve. If the spirals are represented by equation (37) and their centres are a distance apart,
the locus crosses the line o f centres at an angle tan-1 (l/Ape) and passes to infinity on asymptotes
which make an angle cos-1 (27i/?c//) with the line o f centres (figure 5). T he activity is still
indistinguishable from that of one dislocation.
However, if the centres are closer together than half the radial separation between successive
turns, i.e. than 27
Tpcit he spirals have no intersections except near the origin. The
intersections is now an S-shaped curve running from one centre to the other, and no longer
divides the area into two (figure 6). In this case the turns o f both spirals reach the whole o f
the area. In the lim iting case in which the distance between dislocations l is much less than
pc we have effectively the complete Archimedean spiral r = 2
with the branches for
both negative and positive r. Actually it still consists o f a pair o f spirals, which exchange
centres on meeting, at every half-turn. I f
l^ P cthis shift o f ce
T H E G R O W T H O F C R Y STA LS
315
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rotation o f the spirals, so that the activity o f the pair should be twice the activity o f a single
dislocation. For small non-negligible values o f // we m ay crudely estimate that the shift o f
centres imposes a delay corresponding to the tim e required for an unperturbed spiral to turn
through an angle o f the order o f m agnitude ljpci but we make no quantitative estimate
Figure 6. Pair of dislocations of like sign, at a distance
Figure 7. A group of dislocations of the same sign.
beyond saying that the activity o f the pair now lies between 1 and 2 times the activity o f
a single dislocation.
A group of s dislocations of the same
,sig
neach a distance smaller than 2npc from
bour, will generate a spiral system o f s branches. Supposing they are arranged in a line (the
most likely arrangement, since groups o f dislocations usually belong to ‘m osaic’, ‘subgrain’
or ‘lineage’ boundaries), and the length o f the line is , it is easy to see that each branch will
take a time o f the order o f
2 (L-\-2irpc)/v o0to execute a circuit round the group
resultant activity o f the group is sj(l -\-L/27Tpc) times that o f a single dislocation (figure 7).
Vol, 243. A.
42
316
W. K. B U R T O N A N D O T H E R S O N
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I f L is small compared with pc, the activity o f the group is times that o f a single dislocation;
if L is large, and the average distance between dislocations is
L/S, the activity is 27
times that o f a single dislocation.
In the more general case o f a group o f like dislocations not in a straight line, we may
replace L in the above formula by JP, where P is the perimeter o f the group. But in this case
it may happen that the growth fronts have difficulty in penetrating into the group itself.
The dislocation group will still promote growth outside it, but m ay develop a pit in the
surface o f the crystal.
In concluding this subsection we can say that the activity o f a group o f dislocations is in
general greater than that o f a dislocation alone by a factor e, w hich in the case o f a group o f
dislocations o f the same sign can be as great as the number o f dislocations contained in it.
In any case, the distance y 0 between steps produced by the group, far from it, will be given
by their rate o f advance vm) divided by the number o f steps passing a given point per se c .:
e(o/27T. Therefore, using (38),
Vo =
(±
n45)
W e shall see in § 10 that in spite o f the fact that the activity o f a group can be several times
greater than that o f a single dislocation, the absolute value o f the activity cannot surpass a
certain m axim um, the reason being that the rate o f advance decreases when the distance
between steps decreases.
9*2. General case
Suppose now we have any distribution whatever o f dislocations in a crystal face, and a fixed
degree o f supersaturation cr, and consequently a fixed value o f pc (where a and pc m ay be
slowly varying functions o f position on the face). W e now make a formal grouping o f the
dislocations. T he first group consists o f inactive pairs, all pairs o f dislocations o f opposite sign,
closer together than 2
pc.These have no activity by themselves, and their only effe
exceptional cases when they m ay possibly fence o ff a region o f the crystal face, and inhibit
growth there, is to impose a small delay on the passage o f steps originating elsewhere, which
to a first approximation m ay be disregarded. W hen a particular dislocation has two neigh­
bours closer than
2pc,the pairing m ay be m ade arbitrarily, but in such a w ay that
close pairs as possible are assigned to this class. W e now take 2 npc as the effective distance
within which dislocations influence each other’s activity, and by drawing imaginary lines
connecting all dislocations closer together than this, divide all the remaining dislocations
into groups, whose members influence each other’s activity, but in which the groups are
without influence upon each other. The number o f these groups will increase with the
supersaturation. Each group can be assigned a strength
0, ± 1 , ± 2 , ..., according to the
excess o f right-handed over left-handed screws in the group. A group o f strength 0 has an
activity e times that o f a single dislocation, where e is now approximately 1 and generally
slightly greater (the inactive groups o f this strength have already been put into a separate
class). I f the supersaturation is increased, so that decreases, e tends rapidly to 1. W hen it
reaches 1 the group can be subdivided into two, o f strengths s l9 s2, where ^
= 0. I f the
dislocations are in random arrangement, | sx | will seldom exceed 1 or 2, but it must be borne
in mind that dislocations are likely to be in regular arrangements, as in mosaic or subgrain
boundaries, in which case this conclusion does not necessarily follow.
T H E G R O W T H O F C R Y ST A LS
317
pcwill also transform som e o f the inactive pairs into active groups
R eduction o f
strength 0.
Groups o f strength j 4 =0 m ay have an activity up to = | j | times that o f a single dislocation.
W ith increase o f supersaturation, dim inishing
the groups o f strength subdivide into
groups o f strength
sl 9 s2,where ^i+^2 = Since and s2 can be o f opposite signs | s{ \ m
be larger than
s.But w ith increasing supersaturation the groups are ultim ately all sub­
divided into single dislocations behaving independently.
A t every stage prior to this, there is in general some group more active than the rest.
T he resultant activity is always that o f the most active independent group.
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10. Rate of growth from the vapour
Let us first consider the simplest case o f one screw dislocation in an otherwise perfect
crystal surface. T he rate o f growth will then be given by (43), where is the rate o f advance
o f the steps, far from the centre o f the spiral.
T he value o f
vhas been calculated in I, formula (24), that is to say,
<7X„
= 2
vex p ( - W/k T) tanh (
) /?£„(*„, y„),
where a is the supersaturation, xs the m ean displacem ent o f adsorbed molecules, v a frequency
factor, W the evaporation energy,
y 0the distance betw een successive tur
factor taking account o f the fact that perhaps the exchange o f molecules between the step
and the adsorbed layer is not rapid enough to m aintain around them the equilibrium
concentration o f adsorbed molecules, and c0 another factor, w hich is a function both o f y 0
and o f the distance x0 betw een kinks in the steps, and given in general by formula (25), I.
According to the estimates m ade in I for xs and x0 we expect the condition xs x0 to be satisfied
in most cases; then the factor c0 is o f the order o f 1, and the rate o f advance o f the steps is
independent o f their orientation. In this case, using (46) and (44) for the distance y 0 between
successive turns o f the spiral, (43) becomes
R=
where
fiQn0vexp ( —W /kT) tanh (o'1/ff),
<rx — (27
(47)
TPdxs)a = ^rrya/kTxs
(cf. I, equation (30)). For low supersaturations (o'<<icrl) we obtain the parabolic law
R = fiQ.n0v e x p ( - W lk T ) c 2/ ^ .
For high supersaturations
(49)
((r^>cri)(47) becomes the linear law
Rx= j3Qn0(rvexp ( - W /kT ),
(50)
which corresponds to the case w hen xs is larger than the distance between successive turns o f
the spiral. W e see that there is a critical supersaturation alt given by (48), below which the
rate o f growth is essentially parabolic, and above which it is essentially linear. For the typical
values
yjkT~4,
102<z we obtain al ~ 10-1. In figure 8 we plot the factor
R/Ri =
(rif)i tanh (ffi/ff)
as a function o f ff/ffl (continuous curve).
42-2
318
W. K. B U R T O N A N D O T H E R S O N
In the case (an unusual one, we believe) when the condition
expression (47) has to be multiplied by the factor £0(tf/o,1, b) given bY (cf-
+ 2 b(A+B<rl(Tl)tanh
= {1
where
b=
is not satisfied,
formula 25)
x j 2jrxs)
A = In
[4:bxja(l+ (1 + 32)*)],
, b)
0
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0-950
10
0-312
Figure 8. Correcting factor RjRx to the linear rate of growth as a function of cr/crj. The numbers on
the curves indicate the values of
For low supersaturations (a’<|(71) the parabolic law (49) will be multiplied by the factor
+8M ):1;
for high supersaturations
the linear law (50) will be multiplied by
c0(co,b) = (1 + 2Bb)~l,
which can be included in the unknown constant fi. T he factor r0(0, b) is always smaller
than r0(°°> b) . The correcting factor {(cr/crj) tanh (o‘l/a)} r0(<r/o'1, b)/r0(oo, b) is also represented
in figure 8; for a given value o f b, A is also a slowly varying function o f xja. W e see that the
influence o f the factor c0 is important for x0/xs^ l , which we do not think will usually occur.
As x0/xs increases, the linear law is reached for higher values o f crjcr^
Similar considerations apply to the rate o f growth produced by a group o f dislocations. I f
the group is a balanced one (equal number o f right- and left-handed dislocations, strength
s= 0) then there must be another critical supersaturation cr2 below which no growth occurs
/<r)}_1
B=
T H E G R O W T H O F C R Y STA LS
319
at all.
a2will be defined by the condition 2
pairs o f dislocations actually coupled by a step; therefore
p c = /, where / i
ZyajkTl.(51)
<r2 =
For an unbalanced group (strength $4=0), or a balanced group above <r2, the rate o f growth
will be given by
R = Q,n0e(a) v j ^ p ci
(52)
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where the factor e(a) is o f the order o f m agnitude 1 for a balanced group, but can be larger
for an unbalanced group; in both cases e(a) tends ultim ately to 1 when a increases. Using
(46) (assuming
c0 = 1) and (12) for the distance y 0 betw een steps, (52) becomes
R—
fiQ.n0vexp ( —
W /kT )e(a) {or2/o'l) tanh [o'1/e(o’)
W e see from this expression that, however large e(a) is, R cannot surpass the linear law (50).
O n the other hand, R cannot be smaller than (47) for an unbalanced group, but it could
becom e zero for a balanced group below the critical supersaturation <r2.
In the general case o f an arbitrary distribution o f dislocations in the crystal face we expect
to have values for the rate o f growth betw een that for a single dislocation (47) and the linear
law (50), according to the distribution occurring in the particular crystal considered.
A critical supersaturation <r2 o f the type (51) could occur in some random distributions o f
dislocations or a grouping o f dislocations in balanced groups only; nevertheless, there is
evidence showing that the dislocations are distributed in a very irregular way, and groups
o f dislocations o f the same sign occur in mosaic or subgrain boundaries, in w hich case we
do not expect critical supersaturations o f the type (51) to occur.
11. Comparison with experiment
There are few quantitative measurements o f the rate o f growth o f crystals from the
vapour. T he most interesting from our point o f view are those o f Volm er & Schultze (1931).
These authors studied very carefully the growth from the vapour o f naphthalene, white
phosphorus and iodine crystals just below 0° C, under different supersaturations a (from
10“3 to 10-1). For all three substances they found a rate o f growth proportional to the
supersaturation. This linear law was valid for C10H8 and P4 down to the lowest super­
saturation used ( ~ 10“3), but for iodine the rate o f growth becomes smaller than that given
by the linear law when <r<10“2.
Let us first compare their linear law w ith (50) calculated in the preceding section for
high supersaturations. This formula is actually the same as that used by Hertz (1882) and
other authors (Volm er 1939; W yllie 1949) for the growth o f liquids and crystals from the
vapour; it can be written also in the equivalent form
R = pO.p0{2nmk T ) _i a,
(54)
as follows from the equality
n0ve.xp ( —W /kT) = pQ^nmkT)^
(55)
representing the balance between the current o f evaporation and that o f condensation at
equilibrium. /? in (54) is usually called the condensation
, pQis the saturation pressure
and m the mass o f a molecule.
W. K. B U R T O N A N D O T H E R S O N
320
In table 1 we compare the experimental linear laws obtained by Volm er & Schultze (1931)
with (54). The first column gives the average experimental value for R/cr. The supersatura­
tion a was obtained by m aintaining a reservoir at 0° G and cooling the crystal under exam ina­
tion to a temperature —A
°C. Assuming that the vapour pressure between the tw
T
is uniform, the supersaturation at the growing crystal is
The second column
gives the theoretical values for R/jSff from (54), taking for the values measured by Gillespie &
Fraser (1936) for I2, by Centnerszwer (1913) for P4 and by Andrews (1927) for C 10H 8. The
first row gives the Volm er-Schultze results for liquid Hg, for which careful measurement by
Knudsen (1915) showed that fi = 1 , For all the three molecular crystals,
1.
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T able 1. L inear rates of grow th R from the va po u r a t 0° C,
AS FUNCTIONS OF SUPERSATURATION
Hg (liquid)
I2
P4
c 10h 8
R/o" (exp.)
(cm./sec.)
0-66 x 10-6
0-9 xlO - 4
0 9 x 10-5
0-8 xlO "4
Rlficr(theor.)
(cm./sec.)
0-6 x 10-6
3 x 10- 4
0-8 x 10-4
1-5 x 10- 4
A
11
03
0-1
0-5
W
V
(eV)
(sec.-1)
0*66
1013
0-70
0-63
0-79
5 x 1016
1015
1018
The last two columns in table 1 give the values o f W and v deduced from (22). For H g
the frequency factor is o f the order o f the frequency o f atomic vibrations, as we should expect;
in the other cases v is larger, due to the difference in rotational entropy between the crystal
and the vapour.
Coming back to the deviations from the linear law, we notice first that P4 and C 10H 8
follow the linear law to the lowest supersaturations observed. T hat means that
is smaller
than 10-3, and therefore, from (48), that
xs> 104a. This is not
of
xjamade in I, equation (6), is valid for spherical molecules, for w hich the energy o f
evaporation W's of an adsorbed m olecule was assumed to be o f the order o f JIT; in the case o f
a flat molecule like C 10H 8, we expect W's to be larger, and therefore xja will also be larger.
Figure 9 gives the results (logarithmic scale for both axes) obtained by Volm er & Schultze
on several I 2 crystals. T he experimental rates o f growth are not reproducible even for the
same face o f the same crystal; this is not unexpected on the basis o f the present theory.
Assuming that the rate o f advance o f steps is independent o f orientation (*51>#0), one can
choose a value for <
j x~ 0*2 such that most o f the experimental results are contained between
the rate of growth (47) of a single dislocation (continuous line in figure 9) and the linear
law (50) (broken line in figure 9). Taking al ~ 0*2 and y jk T ~ 4, we deduce from (48),
~ 10 2a for 12 at 0°G , which is in reasonable agreement with w hat we should expect
(cf. I, equation (6)).
Nevertheless, one notices that the experimental rates o f growth for the lowest super­
saturations are below the theoretical curve (47); in particular, the rate o f growth at
a= 3-8 x 1 0 -3(A T = 0°*037)
is < 10-3 times the rate o f growth given by the linear law. T he reduction given by formula (47)
is only a factor a/ffl ~ 5 x 10"2 with respect to the linear law. This fact could be explained
in a number o f w ays:
(i)
It is an obvious corollary to our view o f crystal growth that it is susceptible to poisoning
by traces o f impurity, particularly at low supersaturations at which the number o f
T H E G R O W T H O F C R Y STA LS
321
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dislocations producing growth is smallest. From this point o f view new measurements o f
the rate o f growth w ould be very w elcom e.
(ii) There m ay be a critical supersaturation o f the type (51) o f the order o f ~ 10~2, and
therefore the dislocations are about 10_3cm . apart; nevertheless, we are loath to draw this
conclusion from such slight evidence.
(iii) O f course, it w ould be possible to choose a larger value for
in order to explain the
small rates o f growth at the lowest supersaturations, but the corresponding value for ~ 6
*3 10
Figure 9. The rate of growth of I2 crystals at 0° G as a function of <r (Volmer & Schultze 1931) in
a logarithmic scale. The broken line is the Hertz law with a condensation coefficient /? = 0*3.
The continuous curve is the rate of growth of a single dislocation (formula (47)) with
102<z.
The dotted curve is that of a single dislocation assuming *0 = 10, = 103a.
would be too small. Another alternative is to suppose that the condition xs $>>#0 is not satisfied
for I 2. In order to decrease the number o f degrees o f freedom, let us assume that the condensa­
tion coefficient /? is due only to the factor c0(co,b). T hat fixes x0/xs ~ 10. Assuming for
xs ~ 10
2a,one obtains the dotted curve represented in figure 9. As we said before we do not
think that a value x0/xs ~ 10 is actually possible. This point will be decided when the topo­
graphy o f a crystal grown from the vapour is observed. If, as we believe, a:0<^^ the steps must
be circular, if
then they must follow the crystallographic orientation, as is observed
in the case o f growth from solution (Griffin 1950; Frank 1950).
W e have been considering the rate o f growth o f macroscopic surfaces, for which the
growth is due essentially to the molecules condensing from the vapour; in this case formula
(54) represents the m axim um rate o f growth for a given surface. W hen the dimensions o f the
surface are small, diffusion o f molecules from neighbouring surfaces can give an important
contribution to its growth, and the rate o f growth m a y b e substantially greater than (54).
W. K. B U R T O N A N D O T H E R S O N
322
For instance, Volmer & Estermann (1921) studied the growth o f small crystals o f H g at
—63° C. The crystals had a plate shape, the thickness h was not observable but was estimated
to be of the order o f 102a. The rate o f growth o f the edges was estimated to be 103 times that
given by (54). The ratio between the contribution to the growth from diffusion on the flat
surface and directly from the vapour should be 2
Therefore we deduce
105a for
Hg at
T = 210° K , which agrees with the value that we should expect.
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12. Growth from solution
We consider now, very briefly, the application o f the preceding ideas to growth from
solution. Although it is clear that from a qualitative point o f view there is no essential
difference between growth from the vapour and from solution, a quantitative theory o f the
rate o f growth from solution is m uch more difficult.
First o f all, we expect the rate o f advance o f a step in the crystal surface to be a definite
function o f the distance #0 between kinks in the step, because although x0 is always small,
the diffusion o f solute molecules towards the kinks, either through the solution, on the
surface or in the edge o f the step, is now m uch slower than on the free surface o f a crystal. It
is difficult to decide the relative importance o f these three currents. For instance, the ratio
between the current through the solution and on the surface will be represented by the
factor DN 0a/Dsns0, where D and Ds are the diffusion coefficients and
and ns0 the saturation
concentrations in the solution and on the surface respectively; it is likely that D > D S but
also
NQa< n s0, therefore the factor above m ay be greater or smaller than 1. Nevertheless, for
the time being and in order to simplify the problem, we shall suppose that the contributions
from the diffusion on the surface and in the edge can be neglected. Even under these condi­
tions we find another difficulty in the fact that neglect o f the motion o f the sinks is no longer
generally justifiable.
Let us suppose we have a set of parallel steps, at distance y 0 from each other, in one o f the
close-packed crystallographic directions (for which x0, the distance between successive
kinks, is a m axim um ). As an approximation the diffusion through the solution can be broken
up as follows: At distances r< x 0 from each kink we have a hemispherical diffusion field
around each kink (provided D/vkink^>x0, otherwise the m ovem ent o f the kink cannot be
neglected) with the diffusion potential (1 —
a/r)<r(x0)} where <r(x0
a distance
x0;at distances r between #0 and y 0 from each step we have a semi-cylindrica
diffusion, field around each step (provided D/vstep^>y0) with the diffusion potential
[In
(yjx]o"1 O(*o) In (y0/r)+<r(y0) In (r/*0)],
where o’(y0) is the supersaturation at a distance y0; and finally, at distances z from the crystal
surface between y 0 and 8, the thickness o f the unstirred layer at the surface o f the crystal, we
have a plane diffusion field with the diffusion potential [( z —
o’(yo)]
where a is the supersaturation in the stirred solution. The latter diffusion potential applies in
any case, but it would be improper to equate the flux calculated from its gradient to the rate
of growth R o f the crystal unless D/R^>8. U sually Z)/ykink, D/vstep and D/R are not very
large unless both the concentration and the supersaturation in the solutions are small. We
assume in the following considerations that this is the case.
T H E G R O W T H O F C R Y STA LS
323
U nder these conditions and in terms o f the supersaturation <r(*0) the rate o f advance o f
every step is clearly
v* = DN 0Q2 ir<r(x0)lx0.
(56)
As vmis proportional to the num ber o f kinks per cm ., 1/x0, the velocity o f the step will increase
appreciably as the inclination w ith respect to the close-packed slowest direction increases.
Equating the sum o f the hemispherical fluxes going to all the kinks in the step to the semicylindrical flux, and equating also the sum o f the sem i-cylindrical fluxes going to all the
steps in the surface to the plane flux, we elim inate
) and find
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= [l + 2n a (S -y 0)lx 0y 0 + ( 2 alx0)in (y 0lx0) ] - 1,
(57)
which, introduced in (56), gives
vas a function o f #0 and y0.
will clearly inc
increases.
A pplying now formulae (43) and (44) for the rate o f growth R and the distance between
steps y 0 o f the growing pyramid, w hich will be approxim ately valid in our problem also,
we obtain
R =
DN 0Qacr(x0)2(/ 58)
where 2
pc=
2 ya/kTa(x0)is the dim ension o f the critical nucleus and cr(x0) is given by (57).
For low supersaturations the third term in the bracket in (57) is the important one; then
the rate o f growth becomes parabolic. O n the contrary, at high supersaturations, the second
term in (57) is the im portant one, and the rate o f growth becomes linear:
R y=
T he change-over from parabolic to linear occurs at a supersaturation oq, roughly given by
oq
~yxJkT§.(60)
For reasonable values o f y and
8,oq ~ 10-3. A bove oq one should observe on
(59); below oq, all the rates o f growth betw een (58) and (59) could be expected. As far as
we know there is no experim ental evidence for such a critical supersaturation.
O n the other hand, we observe frequently (Bunn 1949; Hum phreys-O wen 1949) that
the rate o f growth is substantially smaller than the linear law (59) would suggest; sometimes
a crystal surface does not grow at all in spite o f the fact that it is in contact with supersatura­
tions as large as
a~Ol. This could possibly be interpreted as being due to the absence o f
dislocations in the surface or to the presence o f so m any that the mean distance between them
is smaller than 2
pc.I n this last case, the number o f dislocations per sq.cm, would have to
be o f the order of 1012cm .“2, which is high. M oreover, in this case, the dislocations would
have to be distributed in a peculiar way, with least density at the centre o f each face; for
otherwise the growth, when it did occur, would be most rapid at the corners, i.e. dendritic.
W e are more inclined to think that the number o f dislocations involved is quite small, .and
that they are situated near the m iddle o f the face. T he changes in growth rate could be due to
rearrangements o f the dislocations or to the effect o f impurities adsorbed on the steps. The
required amount o f such impurity is very small indeed. For example, if the number o f
dislocations per sq.cm, is as high as 108, the number o f atomic sites on the step-lines connecting
them need not exceed 10-4 o f all sites in the area.
Vol. 243. A.
43
324
W. K. B U R T O N A N D O T H E R S O N
EQUILIBRIUM STRUCTURE OF CRYSTAL SURFACES
P art III. S teps and tw o -dimensional nuclei
13. Introduction
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W e begin with an outline o f the theory to be developed in later sections. Some o f the
statements made in this outline receive their fuller justification later.
It is clear that a crystal will grow only if there are steps o f monomolecular height in its
surface, and growth will take place by the advance o f these steps forming new molecular
layers. The rate of advance o f the steps will depend on their structure when in equilibrium
with the vapour; hence the necessity for studying this structure as a preliminary to the study
of crystal growth.
Frenkel (1945) has recently shown that such steps, when in equilibrium at temperatures
above 0° K , will contain a number o f
sc( f. figure 11), i.e. molecul
kin
the energy necessary to take a m olecule from the crystal to the vapour is equal to the evapora­
tion energy W. This is true for intermolecular forces o f a very general character, as has been
shown by Kossel (1927) and Stranski (1928). According to Frenkel, the proportion o f
molecular positions in the step occupied by kinks is given by a formula o f the type
e -w /kT }
(61)
where w is the energy necessary for the formation o f a kink in the step. Our first purpose in
this part is to study in detail the structure o f steps o f any crystallographic direction and to
estimate the value o f w. The concentration o f kinks turns out to be in general considerably
larger than the concentration o f adsorbed molecules in the edge o f the step. W e consider as
a working model a Kossel crystal, a simple cubic structure w ith first and second nearest
neighbour interactions. W hen the crystal is in real equilibrium with its vapour, a step in
equilibrium must be in the mean straight but can have any crystallographic direction.
O n the other hand, if the vapour is supersaturated, it is known that there is a twodimensional nucleus (critical nucleus) which is in unstable equilibrium with the vapour.
Our second purpose in this part is to calculate the shape, the dimensions and the total edge
free energy of the critical nucleus in equilibrium with a given supersaturation at a given
temperature, for the particular case o f a (0 ,0 ,1 ) surface o f a Kossel crystal.
I f one assumes that the crystal is perfect, its growth in a supersaturated environment
requires the formation o f nuclei o f critical size, because it is only when they reach this size
that they are able to grow freely forming a new molecular layer. It can be proved, on thermo­
dynamical grounds, that the number o f critical nuclei created per second must be proportional
to exp (—AJkT), where A0 is half the total edge free energy o f a critical nucleus, which will
also be called activation energy for nucleation (Volm er 1939; Becker & D oring 1935). In the
calculation of^40, the previous authors neglected the configurational entropy, which amounts
to supposing that the shape o f the critical nucleus is the same as it would be at
0° K.
Under this assumption, and for the simple case o f a (0 ,0 ,1 ) surface in a Kossel crystal, the
size o f the critical (square) nucleus and the activation energy A0 for nucleation are given by
kTlnoc = </>ll,
A0In a,
where a is the saturation ratio, defined as the ratio between the actual concentration in the
vapour to the equilibrium value, <j) the energy o f interaction between nearest neighbours and
(6
T H E G R O W T H O F C R Y STA LS
325
l2 the number o f molecules in the critical (square) nucleus. Taking account o f the entropy
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factors, we show that the critical nucleus has essentially the same dimensions given in (62)
but has rounded corners, w hich decreases A0only by a factor o f the order o f 0*8 in a typical case.
W e deduce therefore that the activation energy for nucleation is enormous for the values o f a
for w hich growth is observed ( a ~ 1*01; A J k T ~ 3-6 x 103, for the typical value $ \k T ~ 6),
and consequently the observed growth at low supersaturations cannot be explained on the
basis o f a perfect crystal theory.
W e believe that the observed rates o f growth o f crystals can only be explained by recog­
nizing, as has been suggested by Frank (Burton et al. 1949; Frank 1949), that those crystals
w hich grow are not perfect, and that their lattice imperfections (dislocations) provide steps on
the crystal surface making the two-dim ensional nucleation unnecessary. I f the distance
between a pair o f dislocations producing a step in the surface is such that a critical nucleus
can pass betw een them the step will grow freely. I f that is not the case the step will require
a certain activation energy for growth. U sing our preceding results we calculate this
activation energy, and we show that it is sm all only w hen the distance between dislocations
is practically equal to the size o f the critical nucleus.
In part IV we shall consider the problem o f the equilibrium structure o f a crystal surface
not containing steps, in order to study whether thermal fluctuations are able to produce
steps in the surface, in the same w ay that they produce kinks in a step. T he answer will
be no, provided the temperature is below a certain critical temperature, w hich for the more
close-packed surfaces is o f the order o f or higher than the m elting-point. T he problem o f
the structure o f a crystal surface is actually an exam ple o f a co-operative phenomenon.
14. Equilibrium structure of a step
By a step on a crystal surface w e m ean a connected line such that there is a difference o f
level equal to an interm olecular spacing betw een the two sides o f the line.
I f the crystal lattice contains no dislocations, then there can only be two varieties o f step
in the surface; either the step begins and ends on the boundary o f the surface or it forms
a closed loop on the surface itself (thus bounding a m onom olecular elevation or depression
on the surface). H owever, if dislocations are present, it is possible that a step can start on a
surface and terminate on a boundary, or it can have both ends in the surface. I f a step has
an end in the surface, this end must be a place where a dislocation meets the surface with
a screw com ponent normal to the surface.
For the sake o f sim plicity we shall consider a crystal in contact with its vapour, but many
o f our conclusions will apply for other primary phases; also we shall assume the crystal to
be very large compared w ith the range o f molecular forces involved, that is, we shall speak
of infinite crystals. W e only consider, for simplicity, a (0 ,0 ,1 ) face o f a simple cubic crystal
with forces between molecules o f the nearest neighbour or possibly the nearest and next
nearest neighbour type. Finally, w e neglect altogether the differences in frequency o f
vibration and rotational free energy o f the molecules in different positions in the crystal
surface.
I f a crystal is then in stable equilibrium with its vapour (the vapour being neither super­
saturated nor under-saturated), then it is fairly clear even at this stage (and we prove this
later) that a step in equilibrium in the crystal surface will have a constant mean direction
4 3 -2
326
W. K. B U R T O N A N D O T H E R S O N
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(this direction not necessarily being along a crystallographic axis), and hence under these
conditions no finite closed step can be in equilibrium. The latter will only be in equilibrium
when the vapour is supersaturated or under-saturated. In this case the equilibrium is
unstable. To avoid unnecessary circumlocutions we introduce here the saturation ratio a,
defined as the actual vapour concentration divided by the vapour concentration under
conditions o f stable equilibrium with an infinite crystal surface. Summarizing, then, we
expect to find steps o f constant mean direction (straight steps) if a = 1 and curved steps if
a=fil, the two possibilities being m utually exclusive.
Figure 10. Step at 7 '= 0o K.
Figure 11. Step at T > 0 ° K.
Figure 12. Overhangs in a step.
Potential energy considerations show that at 0° K a step will tend to be as straight as
possible. This is shown explicitly in § 14T (figure 10 ). As the temperature is raised, a number
of kinks appear ( + , —), separated by certain distances (figure 11); a certain number o f
adsorbed molecules [A), and a certain number o f vacant step sites ( ) also appear. A certain
number o f adsorbed molecules (C) also appear on the crystal surface proper. We shall see that
the concentration of adsorbed molecules and vacant sites in the step is small compared with
that of kinks. We require only the knowledge o f the concentration o f kinks to form a picture
o f the structure o f the step. The representation o f the step by kinks, even when we admit
kinks of any height, is not capable as it stands o f including such a feature as that depicted in
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T H E G R O W T H O F C R Y STA LS
327
figure 12, w hich we call a n coverhang ’ ; to this extent our treatment will be slightly inaccurate.
However, when the concentration o f kinks is small, the concentration o f overhangs will be
negligible.
In w hat follows, our unit o f length will always be the intermolecular spacing. W e distin­
guish two types o f kinks, w hich w e call positive and negative, corresponding to a ‘ju m p ’
and a ‘drop’ respectively at the point in question (figure 11). W e use the symbol n+r(x) to
denote the probability that there is a ju m p o f am ount at a point whose co-ordinate is x.
Similarly for n_r(x). W e denote by q(x) the probability that there is no jum p o f any kind at
the point x. In the present case there are no geom etrical constraints, i.e. at each point in
a step, any o f the various possibilities can occur independendy o f what there is at any o f the
other points. H ence the probability o f the occurrence o f a given configuration is equal to the
product o f the probabilities o f occurrence o f the individual situations which make up
the configuration, and we m ay write as a norm alization condition
?(*) +
2
r= 1
K rM +"-r(*)} = !•
(63)
W e define the local m ean direction o f a step at a point x by the equation
oo
h(x) = t a n 0 = 2
r{n+r{.x) ~ n-( 0 < ^ < 1 ),
r= l
where 6 is the smallest angle betw een the step and the [0 ,1 ] direction. Our problem is now
to evaluate the equilibrium values o f q(x), n+r(x) and n_r(x) as a function o f the temperature
T, and also o f the first and second nearest neighbour interaction energies fa and fa.
It is possible to prove in a number o f ways (see appendix C) that the probabilities n and q
must satisfy the therm odynam ical relations
& r(* ) =
(6 5 )
£+ (* )# -(* ) = fi>
(66)
& (*).= & t(0 )a T*,
(67)
where the g represent the relative probabilities
g±r{x) = n ±r{x)lq(x);
( 68 )
we write g± for g±1, and we use the notation
Vi,2= exp ( ~ f a t2/2kT),
W e see, from equation (67), that if a = 1, and therefore the crystal is in thermodynamical
equilibrium with its vapour, all the probabilities are independent o f x, and therefore, from
(64), the step will have a constant m ean direction
O n the contrary, if a =t=1, the step will
be curved; its local m ean direction is then a function h(x) o f position in the step.
14 T .
Equilibrium structure of a straight step. Let us now consider in more detail the structure
o f a straight step. Since there is now no dependence on we shall omit the variable from our
notation. From (63), (65) and (66), using the notation (68), we obtain by summing a
geometric series
(70)
5 + + * - = ---------- ? + ( i - ? ) * I
•
328
W. K. B U R T O N A N D O T H E R S O N
hofthe step, using (64), (6
In the same way, for a given mean direction
g + -g -
(71)
where we have also used (70) to eliminate g++g_.
From equations (66), (70) and (71) we can express , n+ and n_ as functions of
and h. Equations (65) then enable us to find n+r and n_r as functions o f the same quantities,
thus completing the solution o f our problem. The general expressions are complicated
and cumbersome, so we do not give them ; instead, we consider some particular cases.
At T = 0, we get
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q = l —h,
q — 1—2 ^ ,
n_ = 0,
n+ = h,
n±r =
i.e. the step is as straight as it can be.
Before considering higher temperatures, let us see w hat typical values should be assigned
to
faand (f>2’ In our model, the evaporation energy per m olecule is
+
assuming
W ~ 0*7 eV (iodine for instance) and ^2/^i ~ 0*2, we have ^ ! ~ 0 T 5 e V and ^2^0*03 eV .
Thus at temperatures o f the order o f 300° K , we have qx~ 0*05 and rj2 ~ 0-6. Accordingly,
at these temperatures, it is reasonable to assume
1, which amounts to neglecting the
effect o f second nearest neighbours. U nder these conditions the solution to our problem
becomes
(72)
2g+ = l + 7 i - ? ( l - 7 i ) (1 —A),
2g . = l + t j \ - q { l - q \ ) (1+ A ).
For
h = 0, that is to say for the (0,1) step, equations (66), (70) and (71) give
? = ( 1 - ’7i '?!)/(1 + 27 i - m !),
n + = n _ = qq1.
(73)
At low temperatures we get
n++n_ = 2 ^ = 2 ex p ( —
(74)
T he expression n+-f n_ now represents the proportion o f step sites occupied by kinks from
which the evaporation energy into the vapour is W. Thus, we obtain the same formula as
Frenkel (1945) for the concentration o f kinks, the energy o f formation o f those kinks being
here
w=
This value for w is, o f course, equal to the increase in edge energy o f the step
by the formation o f a kink. Since w is small, we shall have a considerable number o f kinks;
in fact, if
~030° K , ^ ~ 0T5 eV , we shall have one kink for every ten molecules in the
T
step. The concentration o f molecules diffusing in the edge o f the step is much smaller than
that o f the kinks, being in fact proportional to rj\ which gives one diffusing molecule per
100 molecules in the step, if we use the above values for
and T.
As the inclination of the step increases
>)0 with respect to
{h
number o f kinks increases, because o f the presence o f kinks due to geometrical reasons;
actually n+ increases and n_ decreases in such a way that n+-{-n_ increases. For very small
(A<2^j), and at low temperatures, one deduces
(7 5 )
T H E G R O W T H O F C R Y STA LS
329
O n the other hand, as we approach the [1 ,1 ] direction
= 1), the positions in the step
from which the evaporation energy is W, are no longer the kinks o f height 1 (probability ),
because o f the influence o f the second nearest neighbour bonds. Consequently the
number o f kinks w ith evaporation energy W reaches a m axim um somewhere near = 1,
and decreases again being a m inim um for the direction [1 ,1 ]. Actually, for the (1,1) step
at low temperatures at w hich rjx and rj2 are small, we obtain
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q = n+2 = rj2 = exp(-<f>2l 2 k T )i
n+ = 1 - 2
(76)
N ow the proportion o f step sites from w hich the evaporation energy is W is represented by
q + n +2. W e obtain again a formula o f Frenkel type w ith an energy o f formation w = \(j)2
w hich is extremely small, m uch smaller than in the case o f the (0 ,1 ) step. A t higher tem­
y 2~ 1, w e obtain
peratures, for w hich
1 1 —7?
I f,
2\
? = 2 l+ ^ ! ’ *+ = iC1 -9f)»
l ~V2l
2
"- = ( 1 + ^ 7 ? ......
/--V
(77)
It can now be seen w hy we have gone to the trouble o f considering second nearest neighbours;
we have done so in order to obtain the correct behaviour at low temperatures for the
directions near (1 ,1 ). In fact for low temperatures (77) becomes
7=
n+ — i> n-r =
n+r = (i)r+1>
independent o f temperature.
14*2. Free energy of steps
It is o f interest to evaluate the configurational free energy o f a straight step. U sing
standard methods
(S = k \n W , F = U —
be arranged in a step) we obtain the following general expression for the edge free energy
per molecule:
P — K ^ id -^ 2 ) + i ^ i + ^ ^ r,(ln ^4-M n^+),
(78)
where, o f course, q and g+ are functions o f h. This expression is referred to the [0 ,1 ] direction;
in order to obtain the free energy per unit length o f the step itself, we must divide the
expression by (1 + A 2)*.
In the general case the extended form o f (78) is exceedingly cumbersome, so we again
consider some particular cases. For the (0 ,1 ) step at low temperatures the formula (78)
gives
^01 =
i^
( 1 + 2^2)—
under the same conditions the corresponding quantity for the (1,1) step is
Fu=
</>l -\-<f>2- 2k T \ n ( l + y 2).8( 0)
T he contribution o f the entropy o f the kinks to the edge free energy is small for the (0,1)
step, but for the (1,1) step it is not negligible. In fact, at temperatures for which
L
formula (80) gives 2 & T ln 2 ~ 0-07, or a third o f <f>1.
At reasonable temperatures the free energy is always smaller for the (0,1) step than for
any other, so we m ight conclude that steps other than the (0,1) steps are not in real equili­
brium, and that there must be a tendency for these steps to change into (0,1) steps. I f the
,T
S etc
W. K. B U R T O N A N D O T H E R S O N
330
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steps are infinite, this conclusion would be erroneous. Actually, every step is in equilibrium
with the same vapour pressure. Moreover, although the (0,1) step has the smallest free
energy, there is no tendency for other steps to change their orientation, because even the
smallest rotation o f step requires the transport o f an infinite amount o f material.
Frenkel (1945) has treated the kinetical problem o f the transformation o f any step into
a (0,1) step, assuming that only the latter are present in equilibrium, on account o f the
higher energy of the former. H e obtains in this way a time o f relaxation independent o f the
length o f the step. This result is clearly incorrect, because the processes allowing a finite
step to change its orientation occur only at the corners; consequently the time required for
this rotation increases with the length o f the step.
15.
The two-dimensional nucleus: activation energy for nucleation
We have seen in § 14 that if the saturation ratio a is unity then the equilibrium steps are
in the m ean straight. I f a 4=1 we shall find that the equilibrium step forms a closed loop;
if ot> 1 the step bounds a finite incom plete layer o f molecules on the surface, in which case
we speak o f a two-dimensional nucleus, and if a < l the step bounds a finite hole in an
infinite incomplete layer.
The most obvious m ethod for calculating the shape o f an equilibrium nucleus would be
to use the free-energy formulae and W ulff’s theorem.* Although the use o f W ulff’s theorem
is simple in principle, the details prove to be cumbersome. However, the results gleaned in
§ 14 enable us to find the shape directly.
W e know that in general (see appendix C)
g±(x) =S ± ( 0 ) a T*,
and to complete the explicit determination o f as a function o f x it is necessary only to
evaluate g (0 ). So far we have not specified an origin for x ; let us now choose it to be the point
where the local m ean direction o f the step is (0 ,1 ). Then, by symmetry,
£+(*) = £ - ( - * ) .
Hence, using (66), we obtain
g +(0) = g_(0) = 7i>
g±
(#) =
so that w ith (67) we get
Invoking once more the normalization condition (63) we obtain
1 —7?tfK2 - ? ! ) + 7 i ( l —7i) (a*+a~*)’
using (65) and (81). It is now clear that our information on the structure o f the step at every
point x is complete.
15 T .
The shape of the equilibrium nucleus
The local mean direction h{x), and therefore the shape qf a step in equilibrium with an
external phase o f saturation ratio a, is clearly obtained from (4), using (21) and (22). In
Cartesian co-ordinates, the shape o f the step is represented by the equation
y = I h(x')
J0
',
* In appendix D we offer a new proof of a generalized Wulff’s theorem, which will be used later in § 16.
T H E G R O W T H O F C R Y STA LS
331
where the origin is at a place where the local mean direction is (0,1) (see figure 13 ). Inte­
grating, we find
yln a ^ l n f t l - M i a * ) (1 —
where
—Vi^l)2} —i I n [//(*) t f ( - * ) /{ t f ( 0 )}2],
( 83 )
H(x)is given by
H(x) = {l-tfftfi(2--7§)}a*-}-7i(l - ? f ) (a2* + l).
The second term in (2 3 ) can be shown to be always small and negative; when
zero.
The value of y in ( 83 ) becomes infinite when x = ± J/', where /' is given by
= 1 it is
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kTlnoc = (^1 + 2^2)//'
( a > l) .
Thus ( 83 ) represents a finger-shaped figure (figure 13 ). It is interesting to note that V
coincides with the dimensions of the equilibrium or critical nucleus when a square shape is
assumed [§ 13 , formula ( 62 )] and
F igure 13. Step in equilibrium with supersaturated vapour
(<^
l k T ~6).
/
If in (83 ) we replace a by its expression as a function of l' from ( 84 ), we see that y//' is
a function only of xjl' ; hence the shape of the step is independent of a. The value of a, fixes
the size of the figure. The shape is, of course, dependent on the temperature; at low tem­
peratures the ‘corners’ become sharper and the ‘edges’ straighter. At temperatures for
which ~ 1 the expression for the shape can be written in the form
y\V =
)af/T
k(n
l {l - [sinh (^ x / 2 £ 77 ')/sinh ( ^ / 4
(85 )
The figure we have obtained is not, of course, entirely correct, in the sense that we should
have obtained a closed figure. The reason is the neglect of ‘overhangs ’ (figure 12), which is
not a good approximation when the inclination of the step is much greater than
with
respect to the (0,1) direction taken as #-axis. Nevertheless, knowledge of the step shape in
a range \ ti enables us to state the complete shape, because of the square symmetry of the
(1,0,0) face of the simple cubic lattice. The diameter l of the nucleus obtained in this way
(see figure 13 ) turns out to be given by
k T In01 = (<f>l + 2^2—
4:kTijx)//,
’
(86)
to the first order in tjv The expression in parentheses is seen to be 2
(equation 79 ), F0l
being the edge free energy per unit length of the (0,1) step. This is the result that we should
Vol. 5243. A.
44
W. K. B U R T O N A N D O T H E R S O N
332
expect from a correct application o f the Gibbs-Thomson formula (see appendix D ), and it
shows that the shape we have calculated is a good approximation. The difference between
l given by (86) and /' given by (84) is practically negligible for ordinary temperatures.
The radius o f curvature p o f the nucleus at the corners when
1, is easily shown to be
g iv en b y
;
/>/^ = —V(2) (*77^0 tanh2
(87)
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The rounding o f the corners at high temperatures is considerable. For the melting-point
TM) for which kTM\<j>x~ 0*6, the nucleus would have practically a circular shape. According
to (87), the nucleus would become square at
T0°K . I f
considered the nucleus would becom e an octagon.
15*2. Activation energyfor two-dimensional nucleation
It is easy to show how the activation energy for nucleation A0 is related to the total edge
free energy F0 o f the critical nucleus. Let n be the number o f molecules contained in a
nucleus, then n will be variable and will have the value nQfor the critical nucleus itself. Let us
suppose that the shape o f the nucleus does not change appreciably for values o f n around n0.
Then the increase in free energy by the formation o f a nucleus containing n molecules will
A= —
where /? is assumed to be a constant given by
F0 = finl
N ow
Ahas to be a maxim um for n — n0, because the critical nucleus is in unstable
equilibrium with the vapour o f saturation ratio
This condition fixes the value o f
n0=
(fi/SkTIn a)2,
and the maxim um o f A is equal to
A0 = n0k T in a =
(88)
T he probability for the formation o f a critical nucleus is then proportional to exp ( —A JkT),
and A0 is called the activation energyfor two-dimensional nucleation.
In the case o f our (0, 0, 1) face in a simple cubic crystal we know that by an application
o f the Gibbs-Thomson formula (see appendix D) the dimension l o f the two-dimensional
critical nucleus is given by
k T ]n a = 2F Jl,
(86)
where F0l is the edge free energy per molecular position in the direction [0, 1], The activation
energy for two-dimensional nucleation can then be written in the form
a _
0
no (2F0i)2
F k T lx ia
Assuming that the critical nucleus is a square o f size l and putting F01 ~
Becker-Doring expression
^ = m rIn a .
(89 )
we deduce the
(62)
Formula (89 ) differs from (62 ) essentially by the factor n0fl2, the ratio of the actual area of the
nucleus to that of the square circumscribed on it. For the typical value (f>\jkT ~ 6 we obtain
graphically from figure 13 that w0//2~ 0*86. This shows that the actual activation for
T H E G R O W T H O F C R Y STA LS
333
two-dimensional nucleation is reduced with respect to ( 62 ) only by a small factor, which is
not sufficient to account for the observed growth rate at low supersaturations.
16 . Steps produced by dislocations
As a final application of the generalized Wulff theorem (see appendix D) we evaluate the
activation energy for nucleation in the presence of screw dislocations.
A real crystal is supposed not to have a perfect lattice, but to contain a number of lattice
imperfections in the form of dislocations, and Frank (1949) has shown that when dislocations
having a screw component normal to the surface terminate in the crystal surface, they
ensure the permanent existence of steps in the surface during growth. Every dislocation is
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B
\
F igure 14. Step between two screw dislocations P
and Q terminating in the surface.
\
/
✓
F igure 15. Equilibrium positions of a step
between two dislocations P and Q: PAQ
stable, PBQ unstable.
the origin of a step, which finishes usually in another dislocation of different sign. Therefore
we must study the behaviour of a step between two screw dislocations of opposite sign.
A picture of the surface in this case is shown in figure 14 . We shall suppose for definiteness
that the line joining the dislocations is in the [0,1] direction, and we let the distance between
the dislocations be d (figure 15 ). In real equilibrium (a = 1) the step will remain straight
between the two dislocations (figure 14 ), having the structure of a piece of step (0,1). If
a > 1 the step will become curved (figure 14 ), and its shape, seen from above, under equili­
brium conditions, is the same as part of the shape of a ‘free’ nucleus passing through the
dislocations P,
Q (figure 15 ). (Our calculations show in fact (appendix C) that the kink
density at a point in a step depends only on conditions in the immediate neighbourhood of
this point.) The diameter of this free nucleus will be /, given by (89 ) or approximately by
( 62 ). If
d,there are two possible equilibrium positions: PAQ which is stable, and PBQ
l>
which is unstable. For growth we require the transition
and the activation
energy Ad for this is half the edge free energy of the piece PBQ minus half the edge free
energy of the piece PAQ. This quantity is easily evaluated graphically. To find the free
energy of a piece of boundary we have merely to evaluate the area of the sector contained by
44.2
W . K. B U R T O N A N D O T H E R S O N
334
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the piece and the lines joining its ends to the centre of the nucleus, as a function of djl (see
appendix D). This we have done for </>ilkT~ 6, and figure 16 shows the ratio of Ad to the
activation energy A0 for ordinary nucleation. The curve has a vertical tangent at l — d,
which means that we have to go to values of l very close to d in order to obtain a reasonably
small value for the activation energy. O f course if
the activation energy is zero. The
dotted curve in figure 16 is obtained if we assume that the free equilibrium nucleus is
a square.
Hence we conclude that given a certain saturation ratio a, and therefore a value l for the
dimensions of the critical nucleus, all the steps connecting dislocations distant d < l will not
move at all, and those connecting dislocations distant
will be able to move freely without
requiring an activation energy.
1
^ 0-5
F ig u r e
dll
16. Activation energy Ad for the growth of a step between
two dislocations a distance d apart.
P art IV . S tructure of a crystal surface as a co- operative phenomenon
17. Introduction
Following an idea put forward by Frenkel (1945), we discussed in part III the structure of
the edge of an incomplete molecular layer on a crystal surface, which we call a ‘step’. We
found that such an edge contains in equilibrium at temperature T a large number o f‘kinks’.
The results of this part enable us to describe the structure at a temperature T of surfaces of
high index, which contain steps even at the absolute zero of temperature. Frenkel appears
also to have concluded that a surface of low index, and thus flat at the absolute zero, would
acquire at a finite temperature a definite number of such steps. This we believe to be incorrect.
We shall show in this part that the problem of the structure of a surface is different from that
of a step, being actually a problem in co-operative phenomena. The result of our investiga­
tions is that a surface of low index will remain flat, and not acquire any steps, below a certain
transition temperature; above this temperature the surface becomes essentially rough,
a large number of steps appearing.
The problem of the structure of a step is actually one-dimensional; we were dealing with
situations spread over a line. As we pointed out in § 14 , we could assign independent
probabilities for the existence of given features (kinks) at each point in the step, i.e. the range
of possible states at each point is independent of the states at all the other points. In the
two-dimensional problem which concerns us, this is no longer the case. This can be seen in
the following way.
T H E G R O W T H O F C R Y STA LS
335
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W e are interested in the difference o f levels betw een neighbouring molecules. I f two
neighbouring molecules differ in level by r molecular spacings, we speak o f dijump o f m agni­
tude r. It is clear that the num ber o f places where jum ps can occur is greater than the
number o f molecules in the surface; hence the probability o f having a jum p between two
given molecules cannot be independent o f the jum ps occurring in all the other positions.
Figure 17 shows a (0, 0, 1) surface o f a simple cubic crystal. A t T > 0 the surface will contain
a certain num ber o f jum ps or differences o f levels. T he levels o f the molecules can o f course
be assigned independently; however, the jum ps cannot. For suppose we trace out a closed
path ABCDEFA in the surface, then the m agnitude o f the jum p at any point on the path
must be uniquely fixed by the m agnitudes o f the jum ps at the rem aining points on the path.
Figure 17
W e shall consider the surface structure problem as solved when the surface potential
energy at equilibrium is known as a function o f temperature. For instance, if we consider
a simple cubic m odel with nearest neighbour interactions
then a com pletely flat (0, 0, 1)
surface has a surface potential energy \<j> per m olecule. If, however, the surface contains
a ju m p o f m agnitude r, then there will be an extra contribution to the surface potential
energy o f am ount
\rp.In general, w e shall define the surface roughness s as {U —U0)
where U0 is the surface potential energy per m olecule o f the flat surface ( = 0) and U that
o f the actual surface; s is equal to the average num ber o f bonds per m olecule parallel to the
surface.
In the case o f a (0, 0, 1) surface in a simple cubic crystal, m entioned above, the problem
o f finding U —U0, w hich we call the configurational potential energy, is seen to be equivalent
to finding the same quantity for a square lattice o f units each capable o f taking a range (i
of states, such that the energy o f interaction betw een two neighbouring units is a certain
function u(ju,ju') o f their states. In the simple case considered we have
=
I
P'\
Our problem, then, is an exam ple o f the so-called standard problem o f co-operative
phenom ena in crystal lattices; given a lattice composed o f identical units, each capable o f
a number o f states ju, such that the energy o f interaction between neighbouring units i, j is
a function of/^- and
p,pwhat is the partition function per unit o f the lattice?
18. Co-operative phenomena in crystal lattices
It seems to be characteristic of co-operative problems that the thermodynamical functions
are non-analytic functions o f the tem perature; they thus possess discontinuities or infinities
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336
W. K. B U R T O N A N D O T H E R S O N
in themselves or in their derivatives. The temperature at which these singularities occur are
called transition temperatures. But this result is by no means generally proved. It is conceivable
that in certain cases there will be a critical region where some of the thermodynamical
functions change very rapidly without having a singularity.
So far, only particular cases of the general co-operative problem in lattices have
been completely solved: those in which p is capable of two values only, and the lattice
is two-dimensional. The partition function is completely known only in the case of
a rectangular lattice with equal or different interactions in the two crystallographic
directions (Onsager 1944; Onsager & Kaufmann 1946). In this case a single singularity is
known to exist and its position is also known. Under the assumption that a single transition
temperature exists for any symmetrical (equal interaction in all directions) twodimensional lattice, its value is known (Wannier 1945), again with the limitation that //is
two-valued.
For our purpose, only the potential energy U is required. There are, in fact, several
approximate methods for finding £/, the best known of which is that due to Bethe (1935).
These methods have been applied extensively (Wannier 1945) to the Ising model of a twodimensional ferromagnet. However, the rigorous solution for a rectangular lattice, due to
Onsager (1944, 1946), is qualitatively different from all the approximate solutions. In the
approximate methods both the potential energy and the specific heat can be discontinuous
functions of temperature; the correct treatment shows that both are continuous, but the
specific heat has a logarithmic infinity at a temperature some 10% below that at which
Bethe’s method predicted a discontinuity. The reasons for this discrepancy have been
discussed by Wannier (1945). These results have thrown considerable doubt on the
predictions of approximate methods, especially when they predict a latent heat, i.e. that the
potential energy is also discontinuous. Broadly speaking, it is characteristic of these approxi­
mations that the calculated quantities are evaluated more accurately on the low-temperature
side of the transition temperature than on the high side.
The exact Onsager solution was applied to the case of a two-dimensional ferromagnet;
the same solution could be applied with little change to an adsorbed monolayer on a perfectly
flat crystal surface. It seems not unreasonable to suppose that the behaviour of a monolayer
will be similar to that of the crystal surface itself. Such an interpretation means that we
suppose that the molecules in the crystal surface are capable of two levels only. This means
that if we include adsorbed molecules or nuclei in the model, holes are excluded. The twolevel model of a crystal surface is undoubtedly an over-simplification, but it has the advantage
that we shall be able to use the results of Onsager’s treatment for several types of symmetrical
and unsymmetrical surface lattices (§ 19 ). The generalization of Onsager’s method to more
than two levels seems to be very difficult; hence, in order to study how the transition
temperature changes with the number of levels, we generalize Bethe’s method to a many
level problem in § 20.
19 . Two-level model of a crystal surface
In appendix E we shall give a short mathematical account of the interpretation of Onsager’s
solution from the point of view of our problem. In this section we shall state the results and
discuss their physical consequences.
T H E G R O W T H O F C R Y STA LS
337
In the case of a two-dimensional lattice with z nearest neighbours and equal interaction
energy
$in all directions (symmetrical case), Wannier ( 1945) has shown that, assuming
there is a transition temperature T0 it will be given by the general formula
gd Hc =
tt/ z ,
Hc = </>/2 kTc,
where gd is the Gudermannian function defined by
gd * = 2 tan - 1ex—\ u = 2 tan- 1 (tanhf#).
In the simplest case of a square lattice (z = 4 ), (91 ) becomes
sinh Hc = 1,
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or
kT
Vc ~ e~Hc — J (2 )
—
0*4 1 ,
(2 In cot ^7r)-1 ~ 0*57.
(92 )
18. The surface roughness
sof a square surface lattice as a function of rj for a two-level model
O, Onsager’s solution; B, Bethe’s solution; I, assuming no geometrical constraints.
In this case Onsager and Onsager & Kaufmann have found the exact partition function of
the lattice for all temperatures. The configurational potential energy per molecule U —U0
or the surface roughness s = 2 (U —U0)/</> (see § 17 ) is found to be given by the formula
(93 )
s = 1 —J | l -\-^k2K \ coth
where
Kx =
K(= f
Jo
is the complete elliptic integral of the first kind, and
k2 = 2 tanh2
1,
kx
[1
sin2
2 sinh H
cosh2 H ’
A graph of
sagainst 7 = exp ( - H ) = exp {—<j>\2kT) is given in figure 18 . The curve possesses
a vertical tangent at the transition temperature given by (92 ). For low temperatures or
small rj} (93 ) becomes
s = 4 4>
(94)
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338
W. K. B U R T O N A N D O T H E R S O N
This result shows that at low temperatures the jumps existing in the surface are due essentially
to adsorbed molecules; in fact, the proportion of molecular positions on the surface occupied
by adsorbed molecules is rj4 = exp ( —
2(f>fkT ) ; as every adsorbed
bonds, the number of these bonds per molecule due to adsorbed molecules turns out to be
equal to (94 ). In an actual crystal surface there will also be vacant surface sites in number
equal to that of adsorbed molecules, therefore s should be 8
the reason why (94 ) is only
half of that is, clearly, the assumption of a two-level model. In fact there will also be on the
crystal surface nuclei of adsorbed molecules and holes consisting of more than one vacant
surface site, but their concentration will be very small at low temperatures. Provided that is
so, they can be considered as independent entities, and their concentration is proportional
to
ji 6for nuclei (or holes) of two adsorbed molecules (or vacant surface sites) and to higher
powers of 7 for greater nuclei (or holes). As the temperature approaches the transition point,
the concentration of these nuclei becomes larger and they cannot be considered as inde­
pendent entities; the problem must then be considered as a co-operative phenomenon.
It is interesting to compare (93 ) with the result that we would have obtained if we had
assumed no geometrical constraints in the surface; it is easy to see that under these conditions
s would have been given by
j = 2 ? /(l> H ),
which is represented by the dotted curve in figure 18 .
In the case of a triangular lattice (
exp 2
or
tjc =
l/*/3 ~ 0 *58 ,
(95)
z = 6, close-packed plane
Hc3,
kTJ(f) = (In 3 )"1~ 0 -9 L
( 96 )
This value for the transition temperature was obtained directly by W annier & Onsager
by an elegant method (W annier 1945) assuming that a single transition temperature
exists.
Examples of square surface lattices are the (1,0,0) face both in simple cubic and facecentred cubic lattices. An example of a triangular surface lattice is the (1,1,1) in the facecentred cubic lattice. In all these cases the nearest neighbour interactions in the surface itself are
the same as the nearest neighbour interactions inside the crystal. The transition temperatures
(92 ) or (96 ) for these surfaces are very high and seem to be of the same order of magnitude
or higher than the melting-point TM of the crystal. In fact, for the solid state of the rare
gases for which a nearest neighbour interaction model can be considered as a reasonable
approximation, we find from the experimental values
0 7 . We deduce that these
surfaces, if the crystal is perfect, must remain essentially flat for all temperatures below
the melting-point, apart, of course, from the presence of adsorbed molecules and vacant
surface sites.
Another interesting case is that of surfaces for which the nearest neighbour interactions
in the surface itself are not only first but also second nearest neighbour bonds, as, for instance,
(1,1,0) both for simple cubic and face-centred cubic lattices. Then the surface lattice is
rectangular and the interactions are
<j)x(first nearest neighbour bon
(second nearest neighbour bond) in the other direction. The exact partition function for this
lattice has also been given by Onsager (1944). He has shown that the potential energy (or our
T H E G R O W T H O F C R Y ST A LS
339
surface roughness) follows a curve similar to figure 18 , with a vertical tangent at a transition
temperature given by
sinh
HXcsinh H2c = 1,
HXc =(j>x\W ro
H2c = $2\2kTc.
This formula can be written in the form
2£ t;
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In coth ^1
4kTc
<h
^2
(97)
Figure 19 gives kTc\<j>x as a function o f the ratio ^2/^i betw een bond energies. W e see that
for a given
<j>xthe transition temperature decreases rather slowly as (j)2 decreases; it is only for
ratios o f the order
$2\<j)
x~ OT that Tc is smaller than (92) by a factor J. This is prob
case for hom opolar crystals. Thus the transition temperature for these surfaces should be o f
the order o f one-half o f the m elting-point. It could be interpreted as a surface m elting o f
second nearest neighbour bonds.
F igure 19. Transition temperature for a rectangular unsymmetrical lattice as
a function of the ratio between the bond energies in both directions.
Finally, in the case o f surfaces containing only second nearest neighbour bonds, as, for
instance, ( 1,1 ,1 ) for simple cubic crystals, the transition temperature (7) is m uch lower; it
is o f the order o f ^2/^i times the m elting-point. A t ordinary temperatures these surfaces
w ould therefore be above their transition temperature.
The configurational surface free energy is, in all cases, very small below and at the transi­
tion temperature. Above this temperature it decreases roughly linearly with rise of tem­
—kIn 2 approximately. Therefore the differences in s
perature, the slope being
energy between the different faces of a crystal will decrease more and more as the temperature
rises and the critical temperatures of the high index surfaces are surpassed, since high index
faces have higher surface energies but lower transition temperatures.
The existence of a transition temperature, when it is below the melting-point, should have
an observable effect on the adsorption properties of the surface, which clearly depends on
the surface roughness. Actually the presence of adsorbable substance will change the
equilibrium structure of the surface itself, and its transition temperature. We hope to treat
this point in detail elsewhere. Meanwhile, a possible experimental method of testing the
existence of a transition temperature would be to prepare, say, a metal crystal with one of
Vol. 243. A.
45
W. K . B U R T O N A N D O T H E R S O N
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340
its less dense-packed surfaces exposed, anneal at various temperatures
, quench and
test the adsorption properties of the surface. A sharp change in the latter would be expected
when the annealing temperature crosses the transition temperature of the surface.
Similar considerations would apply to the influence of the catalytic properties of the surface
of some solids on the kinetics of chemical reactions between adsorbed substances, in the cases
when the catalytic activity is restricted to ‘active’ points on the surface. Frenkel (1945)
suggested that these ‘active’ points should be identified with the presence of jumps on the
surface, assuming that the adsorption of the reacting molecules in the edge of the steps
decreases the activation energy for their chemical reaction.
On the other hand, the existence of a transition temperature will not have any influence
on the kinetics of growth of the crystal surface. It would if the crystal were perfect. Below
the transition temperature the only mechanism of growth would be a two-dimensional
nucleation, which we know (part III) is always a very slow process at low supersaturations.
Above the transition temperature the growth will be proportional to the supersaturation.
Actually the fact that real crystals are imperfect guarantees the presence of the steps required
to explain the observed growth at low supersaturations and below the transition temperature;
Frank (1949) has shown that, during growth, a dislocation or group of dislocations termi­
nating in the surface sends out closed loops of step in such a way that at any instant the
surface is covered by a very high density of steps, practically independently of the number of
dislocations present (provided there is at least one). Therefore, even if we observe the growth
of a surface for which the critical temperature is below the melting-point, there would hardly
be any difference between the rate of growth below and above the critical temperature.
20. Many-level model: Bethels approximation
The two-level model of a crystal surface is clearly an over-simplification. We expect that
with a many-level model, which corresponds to the actual crystal surface, the transition
temperature should be lowered. In order to study this point we extend in this section Bethe’s
method to our many-level problem. In this method we assume that in a given region,
arbitrarily chosen, the probabilities in which we are interested are independent; we then
insert correction factors to take account of the geometrical constraints, and attempt to
evaluate them by the requirement of self-consistency.
We shall limit ourselves, for simplicity, to the study of the structure of a (0,0,1) surface
of a simple cubic crystal. Figure 20 shows a group of five molecules in the crystal surface,
whose levels are i, j ,
k,l, m (at
T = 0 we
probability for this configuration
be p(i;j, k,/, m ) , n o t being norm
p(i ;j, k, /, m)
—7ll~jl + ll_AI +li_*l +l*_mle(j)e(£)e(/)e(m), exp
where ^ is as usual the nearest neighbour interaction. The factors containing rj represent the
Boltzmann factors, and the functions e axe the correction factors which take into account
the influence of geometrical constraints of the outside region on the molecules considered.
The factors
e(x)will be less than unity unless x 0; in this case we take e(0) = 1. The lev
zero corresponds to Bethe’s ‘right atom’, the other levels to different kinds o f‘wrong atoms’.
By symmetry, e(x) =
e(—#).
THE GROW TH OF CRYSTALS
The total probability
o f/, A;, /, m, is
341
p{x),for the central molecule to be at the level *, whatever the values
p (x )=
P{x\j> k, /, m) = {/(*)}4,
2
( 99 )
jklm
where
(100)
j
and the summations are carried out over all possible levels. Clearly/(#) = / ( —#). Following
Bethe, the self-consistency condition is that the probability
for the central molecule to
be at the level x must be equal to that for one of the molecules in the outer shell to be also at
this level. Therefore/?(#) can also be written as
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P(x) —2 Piiix.k, /, m) =*=e(x) 2
i, klm
( 101
i
where we have used ( 98 ) and (100). Hence from ( 99 ) and (101) we obtain
f r(x)^{f(*)Y -
/* (* ) =
i
(102)
Figure 20
The correction factors e(x) have to be determined from the equations (100) and (102).
Now if we divide (102) b
y
f 3(0), use (100), and introduce the functions g(x) =
the equations (102) transform into the system of linear equations
g { x )^ ij]x fle(i) = e ( * ) 2
i
i
which have the only solution
< x)= g(x) = [ f ( x ) m y .
( 103 )
This form of the conditions that e[x) must satisfy is very convenient for numerical calculations.
The potential energy U, and therefore the surface roughness
2 £7/^, can also be written
in a general form. In fact, the energy corresponding to the configuration of figure 20 is
i ~ j I+ 1
|+
| + 1i - m |J,
and therefore the potential energy per molecule U of the surface will be
tj = 1 ^ ^{1
22
i —j I+1 i —k 1+1 i —l | +1 e - m \}p(t;j9 /, m)
yL p {i;ji ki li m)
where the summations are carried out over all values of i, j , k, /, m. Using (98 ) and (100)
the potential energy or the surface roughness can be written in the form
( 105 )
45-2
342
W. K . B U R T O N A N D O T H E R S O N
where the use of the sign of partial differentiation is to indicate that e is to be treated as
constant during the differentiation. Once the e(x) are known from equations ( 103 ) the
surface roughness factor is determined from ( 105 ).
20-1. Two-level problem
Let us first consider, as an introduction, the two-level case, which is the problem initially
considered by Bethe (1935). In this case, from (100)
0,1),
/( 0) —1 -Htf, /(l) =*1+ 7 ;
e(0) = l} *'(1) =
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and formula ( 103 ) becomes Bethe’s equation
e' = ( r £ f / ’
(106)
Figure 21. Bethe’s factors e for the two, three, and five levels.
giving ex as a function of rj. Figure 21 gives (curve 2) as a function of rj. ex is smaller than
1 only for values of rj smaller than 0 -5 . Above
0*5 the only solution of ( 106 ) is
= 1.
The temperature corresponding to
r]= 0*5 has been interpreted as the t
of the corresponding co-operative phenomenon:
rjc= 0 -5 ,
This value is higher than the value A;7 j ^ ~ 0*57 (92 ) given by the correct treatment of
Onsager. On the other hand, the fact that
ei = 1 above
a physical point of view, and shows only that the method is not correct. Actually
1
means, from the point of view of our problem, that the geometrical constraints have dis­
appeared above the transition temperature, and this is obviously impossible. According
to the definition of the factors e, they must be smaller than 1 at all temperatures.
The surface roughness is now given by
s=
4
l + e \ l+ e ^ *
exij 1
( 108 )
T H E G R O W T H O F C R Y STA LS
343
For low temperatures, s = 4 74, the same expression as is given by Onsager’s treatment.
Above rj = 0 -5 , s is given by formula (9 5 ) corresponding to the hypothesis of independency
of the probabilities or
el= 1. The expression ( 108 ) has been represented in figur
comparison with Onsager’s result.
20*2. Three-level problem
The simplest many-level problem which corresponds to the structure of a crystal surface
is the three-level problem. In this case, using (100) (i = 1,0, —1),
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/(0) = 1+ 2^7, /(l) = /(-!) =7+e,(l+72);
= 1, e(l) = e(-l) =
Figure 22. The surface roughness s for a three-level and five-level problem,
as a function of kT/<f>.
The general formula ( 103 ) now gives the equation
_ [-7 + ( l + 7 2)e ,-|3
1 L 1+276] J
( 109 )
for the calculation of
elas a function of 7. The result is represented in figure 21 (cu
In this case
ex —1 is not a solution of ( 109 ) for any finite temperature, as we should expect
on physical grounds. On the other hand, exhas no singularity allowing the definition of a
transition temperature. To decide where this transition temperature is we must wait till s
is known.
The value of s is easily calculated from ( 105 ):
8^
1 + e j7
l + 2ej 1+ 2
( 110 )
T he result is represented in figure 22 as a function o f
the curve has a point o f inflexion
at a temperature given by
tjc — 0 ’45 , kTJp ~ 0 *63 .
(in )
W. K . B U R T O N A N D O T H E R S O N
344
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This temperature can be interpreted as the transition temperature of the three-level problem.
It is substantially lower than the transition temperature corresponding to the two-level Bethe
problem, but still higher than Onsager’s value. The derivative of s with respect to T (which
would correspond to the specific heat in our problem) would have a maximum at this
transition temperature, but not a singularity.
At low temperatures ( 110 ) becomes
s = 8jy4,
which represents the jumps due to the presence in the surface of adsorbed molecules and
vacant surface sites, which were not allowed in a two-level model. At high temperatures s
becomes now larger than 1 (in fact, its maximum value for = 1 is T 8 ). The reason for that
is again that we have now the possibility of jumps of height 2 intermolecular distances, and
therefore the number of bonds parallel to the surface per molecule (equal to s) can now be
greater than 1 .
20*3. Many-level problem
It is interesting to see how the surface roughness behaves with increasing number of levels.
In the case of five levels, one obtains the equations
=
1
(V ±(1
\
± f)
1+
for the calculation of
given by
h +7(1 )f±g2\ 3
2 y e l+
2jy%2
ex— e(l) =
1 + 2 e{+ 2e\
= /ff2+'ff(l +ff2) et + (l + 74) g2\ 3
/
2
\
/
e(—1) and e2 = e(2 ) = e( —2 ). The s
yS1g+7(e? + 2 e2) + (l + 3 y2)e 1e2+ 2 y3ef
1 + 2 rjel + 2 tj2e2
*
,i n
'
The values of
exand e2 are represented in figure 21 as functions of rj and those of in figur
as functions of kTjcj). The curve s has again a point of inflexion, defining the transition
temperature at a value of kTJp which cannot be distinguished from ( 111 ). The inclination
of the tangent increases and hence the height of the maximum for the derivative of We
notice that the difference in behaviour between three and five levels is rather small from the
point of view of the location of the transition temperature. The reason for that is that the new
parameters e which we introduce to represent the new levels are very small in the neighbour­
hood of the critical point.
The calculations can be carried on to any number of levels. In the vicinity of the transition
temperature there is practically no further change. Also the parameters e do not become 1 ,
at any rate for temperatures for which
<
ji *08 , in spite of the fact t
of levels e(#) = 1 is a solution of equations ( 100 ) and ( 102 ).
The change in the value of the transition temperature, according to Bethe’s method,
occurs therefore at the passage from two to three levels. Although we also expect to have
a decrease in the correct transition temperature, when the number of levels is increased, we
do not know whether this decrease will be as substantial as it is in Bethe’s approximation,
owing to the anomalous behaviour of the two-level problem in this approximation.
T H E G R O W T H O F C R Y STA LS
345
APPENDICES
A p pe n d ix A . I nfluence of th e mean distance x q betw een kinks on
THE RATE OF ADVANCE OF STEPS
A l . Single step
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We suppose the kinks to be regularly distributed on the step at distances xQfrom each other
(figure 2 3 ). Let us suppose Ds is independent of direction in the surface; we have to solve
equation ( 14 ) or
(A l)
with the boundary condition
ijr= 0 for
the edge of the step, governed by the diffusion constant De, which is large enough for the
current passing directly from the surface to the kinks to be neglected. Since \jr must be
periodic in x} with period
x0) the required solution of (A l) is
\fr(x, y)
= u
p2 cneTlny cos knx;
l2
L
n
2 7ra/#0
n=o
x;2+ k l
y=±
(A 2)
\y
■X0
i M =i
He::::
X0
1^0-
F igure 23.
Step (y = 0) with kinks at a distance x0 from each other.
T he minus sign corresponds to t/> 0 , and the plus sign to y < 0 ; the coefficients cn have to be
determined. T he current going into one kink
(x= 0 for instan
j = W
W
) h
going from the surface to the edge integrated betw een — \x 0 and %x0. The velocity o f the step
will then be the result o f the current going into all the kinks, and turns out to be
= 2
<rxsv exp ( —
) fic0,
(A 3 )
which is the general expression ( 19 ) given in § 4 . We have now to calculate the factor c0.
In the edge of the step we shall have a supersaturation cre and an expression i]re —
cre,
which will be a function of x. In general, f(x, 0) —
where ^ < 1 is a retarding
factor similar to ( 17 ). If the interchange of molecules between the edge and its immediate
neighbourhood is rapid,
= 1. Hence
oo
n=0
M e ( x )=
2
(A 4 )
W. K. B U R T O N A N D O T H E R S O N
346
Near the kinks (# = 0 for instance) we shall have f e(Q) = /?20", where /?2 is another retarding
factor corresponding to the interchange between edge and kink. Putting /? =
we have
the condition
oo
2 ^ = 1.
(A 6)
The current in the edge passing through the point A (figure 23), whose co-ordinate is
is given by
»
St=
- D e(dnjdx)= - D ene0a
n=
0
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where (A 4) has been used. O n the other hand, this current must be equal to that going from
the surface to the edge between x and
that is to say
rixo
^DsnsA
Jx
r
(tyldy)y=Odx = - 2 Dsns o r f \ Co{iXoSgnX-x)lxsL
where sgn
xsi 1 for a? > 0 , — 1 for # < 0 and 0 for
the coefficients rn(« > 0) as functions o f c0:
00
2 M n/*n)sinM
n- 1
~1
J
> (A 7)
0. Equating (A 6) and (A 7) we obtain
cjo= (^bc/ri2) [1 + (2c/n) (l'+ # 2/»2)*]-1,
with the abbreviations
b=
x0/2 ttxs,c =
x0al27TX2ei
x2e = DeneQa
From (A 5) and (A 8) we obtain
llc0 = 1 + 4
be2 {^2[1 + (2cIn) (1 + # 2/w2)*]}-1.
W=1
(A
T he series in (A 10) can be approximated to by an integral
1 w
+(*/ *> a
- j;
which can be evaluated using the transformation tan # = bx. T he final result for c0 is
— — 1 + 2 bc(b2-\-c2)~i { f 1(xli-um) " t^ (% « OT)}j
(A ll)
co
where
/,(*!,«„) = 2 ( 4 - I j - 't a ir 1
l)‘/ ( x i- a m)}.
/ 2(*2>»«.) = (1 -4)-*ln{(«„(l + (1 -*1)‘) +x2) l ( u j l - (1-*!)') +*2)},
um= A/{1 + (1 +42)'},
=c/A+(c2/42+ l ) i > l ,
*2 = -c/4 + (c2/i2+ l)*< l.
Foi* x0->oo, we obtain the result for widely separated kinks; it is easy to see that c0 is then
proportional to l/*0, or to the number of kinks per cm.
T he method used to calculate the current going into the kinks is correct, provided the
current via the edge is important or xe> a. In the lim iting case when xe ~ a the result o f this
calculation should be the same as if the current via the edge were neglected altogether. In
§ 4 we have seen that xe is o f the order o f
a ;under these conditio
xs^>a, one obtains from (A 9) that
1 and
cjb^1; then, th
(A ll) reduces to
l/c0 = l + 24]n{4c/(l + ( l + 6 2)*)},
which is the formula (21) of § 4.
(Al2)
T H E G R O W T H O F C R Y STA LS
347
A 2. Parallel sequence of steps
W e suppose the steps and kinks are disposed as in figure 24, the kinks forming a rectangular
lattice. Let the distance betw een kinks be r0 and that betw een steps yQ. Between steps the
sam e continuity equation (A l) holds. A gain w e have periodicity in the ^-direction, hence
in the interval
i."% ..^
i„
i„ ^ v^ i v
cosh lny
f { x>y)
we have the solution
~ 2 cn— i, 1/(A 13)
cos knxy
n= 0
cosn 2^ y 0
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where kn and ln have the same significance as in (A 2). T he velocity o f every step will be the
result o f the current going into every kink, equal to the current going from the surface to the
step in the range r0. T he velocity turns out to be
tfoo = 2 w s ve~ wlkT tanh ( y j
(A 14)
which is the general formula (24) given in § 5. T o calculate c0 we have again the condition
(A 15)
i t , = i
71= 0
as before. T o evaluate all the cn(n>0) as functions o f c0, w e use the same m ethod as in § A l ,
and from (A 15) we deduce for c0 the rather com plicated expression
1 jc0= 1 +
4tbcanh
t
(y0/2 r j J
{n2[l + (2c/ri)(1 + ^ 2/w2)i tanh ((rc/2*)
71=1
where b and c are given by (A 9) and
e=
x0l2
y
(A 17)
_
_!___
J
1
1
A
\Vo
• jtffe
\
.
K
rr
. 7 -----
1
F ig u r e 2 4 .
Sequence of steps distant y0 from each other with kinks distant *0.
In the lim iting case when xe ~
mately evaluated, replacing the ta n h r by
,a and therefore c^> 1, the series in (A 1
x/(l + # ) . T he approxi
/c0 = l + 2^ tan h (y0/2r0) [ln (4 r /(l + (14-^2)i) ) + ( 2 ^ ) t a n -1 ^],
1
which is the formula (25) given in § 5.
A p p e n d ix
B. T
h e m u t u a l i n f l u e n c e o f a p a i r o f g r o w t h s p ir a l s
W e now reconsider, in more detail, the interaction o f the growth spirals o f a pair o f
dislocations (like or unlike) whose separation is at least a moderate multiple o f
W e have
seen (§ 9T) that their resultant activity will equal that o f one dislocation except in so far as
Vol. 243. a .
46
(A18)
W. K. B U R T O N A N D O T H E R S O N
348
influences transmitted along the step from the point of meeting modify the rate of rotation
of the separate spirals. To examine this effect we first consider a simpler case.
A circular expanding step, whose position is defined by = r/J), is helped on over a small
portion of its length, e.g. by meeting another small closed loop of step (a small island nucleus).
Re-entrant portions of the curve fill in rapidly, so that we then have r =
+ / # , t), where
6 is the angular polar co-ordinate and t the time. We suppose/small compared with rl9 and
drftO = r' small enough for its square to be neglected in expressions for curvature (35) and
normal velocity (36):
Ip= (r2+
1
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v= (
r'2—rr")
2
dr/t)r{r2-\-r'2)~* ~ drjdt — drlldt-\-dfldt.
Here s is the arc distance rd. By equation (34) we have then
drjdt+ df/dt = vK- v ^ p j^ + v ^ p ^ f/d s2.
Subtracting the corresponding equation obtained when f is zero, we have
df
It =
d2f
This is simply the diffusion equation, with an effective diffusivity v^pc. W hatever the initial
form of / , provided it is confined to a small portion o f the circumference and disregarding the
closed nature o f the curve (as is right, since we are going to apply the result to a spiral instead
o f a circle), its solution tends rapidly to the form
A(4:7rpcviat)~i exp ( —s2l4:pcv0
/=
/» 00
fd s has the constant value A. Thus the growth increm ent remains constant in
for which
J
—00
area but gradually spreads out along the step.
This result should be approximately valid for deformations o f the spiral also, except close
to the centre, where
/is no longer negligible. W e apply it, then, to a growth spir
once in every turn, meets another based on a dislocation a distance l away. Each time this
happens the resulting concave region o f the growth front fills up rapidly, making an area o f
increment which we estimate roughly as (4 —
u)/2/8 from the
circular quadrants and a rectangle. This occurs (o/27r times a second at a distance, measured
along the step, approximately l2/lQpc from the dislocation. These growth increments now
diffuse along the step, but at the same time the spiral continues to rotate, so that while the
increment spreads, its centre recedes from the dislocation, the arc distance being expressible
approximately as
s = PcVIfyc+vt)2.
Near the centre of the spiral the concept o f an area diffusing along a line fails; the growth
increment which diffuses into the centre is used up in extending the step line faster than would
occur spontaneously.
Supposing the diffusion law continued to hold as far as the centre o f the spiral and on
some fictitious line beyond it, the value o f
fat this point result
growth fronts at a time
t= 0 would be approximately
f d = {(4
tj)l2llQ(7rpcvV)t)i}exp
•
T H E G R O W T H O F C R Y ST A LS
349
This is zero for small or large t, and has a m axim um very nearly at the m axim um o f the
exponential factor w hich occurs at t = //12<yyOc. Inserting this value o f t, and at the same time
writing o) =
e v j 2 p c}w here
esi a factor representing the increa
o f an unperturbed single spiral, w e have
S i, max. = [ ( 4 - * ) / 1 6 ] (6e/3/jr/>c)‘ e x p [ — (£e) (//3/>c)3].
W e m ay alternatively focus attention on the area o f increm ental growth which would have
diffused past the centre, on the same assumptions. This is
Fd =
l 21 IQ
] refc
[ ( 4 —tt)
[(//4/>c+&tf)2 (4t;«, *//> ,) ~ * ] ,
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where erfc (
x)is the com plem entary error function I —erf (#). As a function o f time this
has its m axim um value at
t=
lfl2(opc)and is then
Fd,max. = [ ( 4 - tt) /2/16] erfc [(J»)» (#/»/>,)*].
Consideration o f either o f these expressions, f d max> or Fd>^ x , suffices to show that the
influence transmitted into the centre is quite negligible if / m uch exceeds 3
For variations o f
IIpcthe first o f these functions is a m axim um precisely, the second
nearly, w hen (/e*/3/?c) = 1. T he corresponding m axim um values are 0*234 and 0*155
W e m ay crudely estimate the order o f m agnitude o f the am ount o f extra rotation produced
by such an increm ent by dividing by 27ipc in the first case or up2in the second,'obtaining 0*037
or 0*049e_s o f a turn. This occurs once in every turn, i.e. e ~ T 04 or 1*05. This result is not
directly valuable, for the m axim a occur at a separation too small for the validity o f approxi­
mations used in the treatm ent; but it does show that at greater separations where the
approxim ations are reasonably valid (say l > 47
the interaction is quite negligible.
T he calculation affirms the surmise that w hen pc is steadily reduced below the critical
value \ l above w hich the activity o f an unlike pair o f dislocations is zero, the activity first
rises above that o f a single dislocation before settling dow n to equality w ith it; but it leaves
m uch doubt as to the actual m agnitude o f the m axim um excess. It is probably a few units
per cent, and if it later turns out that im portance attaches to the actual value, a step-by-step
trajectory calculation must be carried out.
A p p e n d ix
C. P r o o f
o f c e r t a i n f o r m u l a e i n t h e s t a t is t ic s o f k in k s
T o prove formulae (65), (66) and (67) in the text, we consider particular processes
(figures 25, 26, 27, 28) and we apply the principle o f detailed balancing.
F igure 25
F igure 26
□
1 • - ■l
F igure 27
-—
3— l h — i
F igure 28
46-2
350
W. K. B U R T O N A N D O T H E R S O N
Let us first consider the process shown in figure 2 5 . The molecule denoted by a square is
the one which moves. The positions in the shaded regions are supposed to remain the same
during the process. In this case the energies of the two configurations are the same, so the
probabilities for their occurrence are equal. Hence
w+rW ? ( * + ! ) w+(* + 2 ) = «+(,._!)(#) »+2( * + l ) q {x + 2).
(C l)
Here we have written n+ for n+l. Later we write g+ for g +1. It is convenient to introduce a
g±rW = n^x)/g(x).
function defined by
Then (A l) can be written
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g + r (* )g + (* + 2 ) =
(C 2)
g + (r -l> (* )g + 2 ( * + ! ) •
For the process o f figure 26, w e need to supply an energy
right-hand diagram. Hence
(C 3 )
in going from the left-hand to the
g+2(x + l) = g + (x )g +(x + 2 )ril
(C4)
where we have again used (C 2) and also the abbreviation
7i,2 = exP ( - ^ i ,2 ftk T ).
(C5)
Substituting the value (C4) o f g +2( # + l ) in (C3) we obtain
(c«)
as is easily shown by induction on r. Formula (C6) is formula (65) o f the text.
W e now need an equation relating neighbouring positions. Comparing (C4) and (C 6)
(r = 2) we obtain, imm ediately,
{g+ M P = •? + (* -!) £ + (* + !)•
From the process illustrated by figure 27 we obtain
g+(*+1)
(C 7 )
’l l =
g+ (*)
g -(*+.!)
and using the property (C7) we obtain
£ + (* )« -(* ) = fi,
which is formula (66) in the text.
The general solution o f the functional equation (C7) is easily proved to be
g + (x )= g +(0) ^ ,
(C 8)
(C9)
where ^+(0)and c are arbitrary constants. W e now investigate the dependence o f the
constant c in (C9) on the supersaturation, or more specifically the saturation ratio a defined
by the equation
N = aN0 =
(CIO)
Here N is the occupation probability for an adsorbed molecule on the surface, and
N o=
e x p { - 2 (^i
is clearly the value of JVwhen
a= 1. Since we are speaking of equilibrium
of position in the surface. From the process represented by figure 28 we obtain
-ixl) ,
W
g+
^
N
( G il)
and combining this result with (C 8), (C9) and (CIO) we obtain
a = e~c,
and therefore (A 9 ) becomes
which is equation ( 67 ) giyen in the text.
g+(x)= g+(0 ) a~x,
T H E G R O W T H O F C R Y ST A LS
A p p e n d ix
D. W
351
u l f f ’s t h e o r e m
4In a crystal at equilibrium, the distances o f the faces from the centre o f the crystal are
proportional to their surface free energies per unit area. 5
A great deal has been written on the general form o f this theorem, and von Laue ( 1943)
has given a critical review o f the subject. There is no really satisfactory proof o f the threedim ensional W ulff theorem even now. T he proof for the two-dimensional case given here is
believed to have merits not to be found in any earlier proof.
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T he problem is to find the relation betw een the shape o f the two-dim ensional equilibrium
nucleus, and the polar diagram o f the free energy o f a boundary elem ent as a function o f its
orientation. T he shape is fixed by the condition o f m inim um total free energy for a given
area o f nucleus. Let (r, <j>) be the polar co-ordinates (figure 29) o f a point T o f the crystal
boundary
Sand
,
let (x,y) be the corresponding Cartesian co-ordinates. Construct a tangent
to S at T, and let OM be a perpendicular from the origin to the tangent (length p ) . Let
f{6) be the edge free energy per unit length o f the elem ent o f boundary at T. The line
elem ent for the boundary in parametric form is
ds = (#2+
y2Y (dt,D l)
where the dot denotes differentiation w ith respect to the parameter
W e now choose 6
as the parameter. T hen the total edge free energy F and area n0 (number o f molecules) o f
our nucleus are given by
p =
respectively. From figure 29,
xcosd-\-ys(D 3)
Let us find the locus o f
Ms aT goes over the w hole o f the curve S. This will give the ‘p
o f S, which is determ ined by (D 3) and the equation obtained from it by partial differentiation
w ith respect to 6. Conversely, if the pedal is known, i.e. p is given as a function o f then the
(x,y) equation o f S can be obtained from the equations
x=
y
cos 0 —p sin 0, j
(D 4)
/>sin#+j&cos0.j
U sing these expressions we can write (D 2) in the form
(D 5 )
T he problem now is to m inim ize F subject to the condition o f n0 being constant. Intro­
ducing the Lagrange multiplier A, the appropriate Euler equation giving the minimum
condition is
- n
dp
d e x d p j^ d d 2\dp)
where
These two equations reduce to
5
Q=
p +p = A (/+ /),
and the solution o f this differential equation is
p(0)=Csm(6-p)+Xf(6),
(D 6)
352
W. K . B U R T O N A N D O T H E R S O N
where C and /? are arbitrary constants. Now the first term on the right possesses the period 2 ;
so that if we choose the origin in such a way that the crystal possesses some rotational
symmetry with respect to it (centre of the crystal), we see that
0. Hence
= m
m
(D ?)
which shows that up to a constant factor the polar diagram o f the edge free energy is the
pedal o f the equilibrium shape o f the crystal.
The foregoing theorem, which is a generalization of Wulff’s theorem, enables us to study
the properties of the critical nucleus in a very general way. First we deduce from (D 5 ),
(D6) and (D 7 ) the total edge free energy F0 of our nucleus:
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F0= 2/z/A.
T he constant A can be determined in the following way. Assuming that the shape o f the
nucleus does not change in the neighbourhood o f the equilibrium dimensions, it was shown
in § 15*2 (formula 88) that the total edge free energy o f the critical nucleus is given by
F0 = 2 n0k T In a.
F ig u r e
Therefore, from ( D 8) and (D 9 ),
and (D 7) becomes
(D 9 )
29
kTin a,
1/A =
kT \noc=f(6)lp(6). ( D 10)
This equation represents not only a generalization o f W ulff’s theorem, but also a generaliza­
tion o f the Gibbs-Thomson formula— in two dimensions o f course. K n o w in g /(0 ), equation
(DlO) gives p(6), and hence the shape o f the critical nucleus is derived by the following
geometrical construction. Draw a radius vector from the origin to the curve o f p{6) and
construct a perpendicular to the radius vector at the point o f intersection with the curve.
Then the envelope o f these perpendiculars, when the radius vector describes a complete
revolution, defines the shape o f the critical nucleus. Naturally, the curve o f p{6) will be
closed if the free energy per unit length is single-valued. W e emphasize here that 6 is generally
not the same as 0, the polar angle in the shape diagram (figure 29).
For the particular points of the step for which the tangent to the shape is normal to the
radius vector, 0 = <j) and/>(#) = r(^), and equation (DlO) becomes
hT\
(Dll)
T H E G R O W T H O F C R Y ST A LS
353
showing that the distance from the centre to one o f these particular points is proportional to
the edge free energy per unit length at this point, w hich is the ordinary form o f W ulff’s
theorem. I f / o r
rsi independent o f (j>then ( D l l ) becomes the G ibbs-T hom son equation in
two dimensions.
It is possible to give a sim ple expression for the radius o f curvature o f the nucleus at a
point whose polar angle is (j>in terms o f the values o f f(6) a n d / / ) at the corresponding point
in the edge free-energy diagram . T he result is
Loa,
( D 12)
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the unit o f length again being the interm olecular spacing.
T he solution (D 7) has been obtained on the assumption that
and its pedal possess
continuously turning tangents. W hen there are sharp corners in S, the solution also
applies for all pieces o f S possessing continuously turning tangents. H owever, it m ay turn
out that more than one free-energy diagram corresponds to a given
T o illustrate this, let
us consider the polygonal equilibrium shape shown in figure 30, where S is the boundary
o f the crystal and 0 is its centre.
P1
F ig u r e
30
It is easily seen geom etrically that if the free-energy diagram is that figure P obtained by
finding the pedal o f
Sa( number o f circular arcs, if S is polygonal), then any curve P
entirely outside P, but coinciding w ith P at the cusps C, w ill give the same S, nam ely that
given by P. If, however, the free-energy diagram lies entirely inside P, but coincides with P
at the cusps C, then we obtain a shape for the crystal S' (different from S) which possesses
neither sharp corners nor straight edges, and there is a one-to-one correspondence between
S' and the free-energy diagram, so that given S' the free-energy diagram is determined
uniquely and vice versa. This is the case in the particular model which we have been
considering in part III. In the interm ediate cases when the free-energy diagram P" lies
partly within and partly without P, then the corresponding crystal shape S" will have sharp
corners with or without straight segments in the boundary, depending on the actual form
A t T = 0 we expect most crystals to be polygonal (or polyhedral, in three dimensions).
T he question arises: Are the corners rounded when
0, or can it happen that the equili­
brium form remains polyhedral ? I f the potential energy were like P' in figure 30, and if the
entropy correction were insufficient to bring the free-energy diagram within P, then sharp
W. K. B U R T O N A N D O T H E R S O N
354
corners would remain. This seems to be the case for ionic crystals. Shuttleworth (1949) has
calculated the potential energies per unit area o f the (1, 1, 0) and ( 1, 0, 0) faces o f a number o f
ionic crystals and finds that their ratio is always greater than 2. Thus it would require
temperatures probably above the m elting-point to give rounded edges. In the case o f metals
near their melting-point, it is probable that the free-energy diagram w ill be within P,
therefore the edges will become rounded.
A p p e n d ix
E.
A n o u t l in e o f t h e m a t r ix m e t h o d o f t r e a t in g
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C O -O P E R A T IV E P R O B L E M S
Here we give a brief account of the mathematics leading to (93 ). The general methods are
due to Montroll (1941), Kramers & Wannier (1941), Onsager (1944), Onsager & Kaufmann
(1946) and Wannier (1945). We consider a chain of identical units, each capable of a range
of states ju. Let the chain be
m + 1units long, so that there are m bonds c
(figure 31 ). Let the state of the rth unit be /
ir. If the units are
contributes u(/ir+l) fir) to the total energy of the chain, so that the total energy of the chain is
=
+
(El)
Hence the partition function for the chain is
/m+l = 2 « P
i
= 2
I • • • ! 2 E(Am+i> / 0
Pm +\ Pm
where
P2
...
(E2)
ft1
T(/*,/) = exp {—«(/*,/)/£ 7 },
(E 3 )
and the summations in (E2) are carried out over all possible values of all the /*’s. Since (E2)
is in the form of a matrix product, (E 3 ) can be regarded as a matrix, /i being a row index,
and n' a column index.
0 - 0 - ....... ~ ( D ~ ©
F ig u r e
31
If we define
= 2
Pm
wc see from (E2) that
^
2 — 2 E(Ah-1 > /0
Pm-\
p \
^
= 2
(E 4 )
(E5)
Am
<J>m+1 may be called the partial partition function relative to the state //OT+1, and we see that
(E5) gives the effect on the partial partition function o f a chain by the addition o f another
unit. The complete partition function
fm
lis o f course given by
+
fm+1 ==: 2 ^m+l^m+l)*
P m +\
If the chain is very long, the ratios between the various components of
will be the same
as those between the corresponding components o f
so that in the lim it as m becomes
very large we obtain
(E 7)
THE GROWTH OF CRYSTALS
355
f m+l = A/m,
(E 8)
H ence from (E 6)
so that A must be the partition function per unit. C om bining (E 5), (E 6) and (E 7) we are
confronted by the following eigenvalue problem for A:
fit*’)
2
fl’
(E 9)
w hich can be written in the contracted notation
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(y > f) (/< )= •¥ (/< )•
(E io )
By (E 3) the elem ents o f V are all positive, and by (E 4) the components o f f (i.e. the
com ponents o f O) are all positive. T his enables us to conclude by theorems due to Frobenius
( 1908, 1909) that A is the largest eigenvalue o f V. T he results used here are that a matrix with
positive elements has a largest eigenvalue w hich is sim ple and greater in absolute m agnitude
than any other eigenvalue, and that, moreover, the eigenvector belonging to it has
com ponents o f only one sign. W e m ay choose this sign to be positive. This eigenvector is the
only one w ith this property.
E l . The
sal, two-level case
e-dim
on
As a preliminary to the solution o f our tw o-dim ensional, tw o-level problem , we first find
the V for the corresponding one-dim ensional problem . In this problem the interaction
energy betw een two neighbouring m olecules is zero if they have the same level and \(j)x if
not. W e designate one o f the possible levels by
n= + 1 and th
a ction 5 energy betw een two neighbours can then be put in the form
(E li)
w hich gives 0 if
where
n= / / and \(f)Yotherwise. H ence from (E 3)
= exp {\H (pn'—1)},
(E 12)
H — ^ j^ kT .
(E l 3)
T he operator F in ( E 12) has the following effect on a general function (which we may
interpret, if w e please, as the partial partition fu n ction ):
(7, f ) M = f ( / ' ) + « ' Hf ( - / < ) .
(E14)
using the contracted notation. I f we define an operator C by the equation
we can write V as
using (E l4 ). From (E l5 )
(C ,t) (/0 = = jK -7<),
V= 1
+e~HC,
C2 = 1,
(E15)
(E 16)
(E ll)
so
Cas
h the eigenvalues ± 1 , showing that F h a s the eigenvalues 1
H. T he upper sign
gives the largest eigenvalue, and hence the partition function per molecule.
In order to use these results in the treatm ent o f the two-dimensional problem, we write V
V = A e i«c,
in the form
(E 18)
w hich is possible in view o f ( E 17), since we have, using (E 17) and (E 18),
V — A (cosh \H + C sinh \H ) .
Vol. 243.
A.
47
356
W. K. BURTON AND OTHERS ON
A2= l - e ~ 2H = 2tf-H sinh
Thus, from ( E l 6),
(E 19)
coth \H = eH.
V—
Hence
where
(2e~H
s inh
yex p
H
(\HC),
(E21)
H
s igiven by (E 20).
E 2.
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(E20)
The two-dimensional, two-level case: rectangular lattice
In order to treat the two-dimensional case we consider n parallel chains o f the same type
as before. W e shall build up the two-dimensional lattice in two steps by the addition o f
complete columns o f molecules. Thus our ‘u n it’ is the colum n. In the first stage we include
only the ‘horizontal’ bonds (left-hand, figure 32), and in the second stage we insert the
‘vertical’ bonds (right-hand, figure 32). W e associate with the last elem ent in each chain
(thejth chain) the variable ju, which can take the values ± 1. A com plete set o f values assigned
I
—x
I
I
—m—' »o
I
I
—x —— x —- - o
F ig u r e 3 2
to the n variables describes a configuration (fi) = (/q, ...,/<„) o f the last column, which
constitutes our unit in the sense o f figure 31. Thus the operator which describes the addition
o f a new column with the horizontal bonds only is
Vx = (20_/r s in h //)in exp ( \ H B )
(E22)
B= Y Ch
from (E 21), where
j= i
jhave the effect
C
and the individual operators
(^jf» 1^) (/fi>
= ?^( / f i j , , , >
Mj) • • •>fin)'
(E 2 4 )
W e now wish to find the operator which represents the insertion o f the vertical interactions,
as in the right-hand figure (figure 32). T he total energy corresponding to the inclusion o f
the vertical bonds will be
........(E 2 ®)
where %<f>2 is the strength o f the vertical bond when it is not zero. W e assume for generality
that
iand (j>2 are not necessarily equal. The effect o f the interaction is to multiply the
(f>
general term in the partial partition function, represented by one o f the 2” vector com­
ponents ^(/q, ...,/<„) by the appropriate factor
THE GROWTH OF CRYSTALS
357
T he corresponding operator is represented by a diagonal matrix. It can be constructed from
the simple operators Sl i ...,S n w hich m ultiply f by the sign o f its first,
wth argument as
follow s:
(E26)
{S j,f) (/»1>—>/</>—> /0
A=
(E27)
i =1
V2 = exp { —
i(n —1)
'}exp [\H 'A(E28)
H
),
H' = $ j2 k T .
where
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H ence the operator associated w ith our problem is
V= V2VX= exp ( —
)ex p { - £ ( n - l ) //'} (2 sinh//)*" exp (\H'A) exp [\HB).
H
\n
(E29)
T he largest eigenvalue o f V is then the partition function per colum n o f the lattice, and its
nth root is the partition function per m olecule.
Onsager ( 1944) has shown that the largest eigenvalue AM o f exp {H'A) exp (HB) is given
by
2 lnA^ = y (
2
n/tt)
+ y (iitjn ) + . . . + y ( 2n) +
where |
c<| / / ' and
H sinh 2
cosh y{(o) = cosh 2 / / ' cosh 2
This leads to the conclusion that, in our case, the partition function per m olecule A is given
by
lnA = - F / A r = £ l n ( 2 s in h t f ) - £ ( # + # ' ) + ; i- P /(« )(/« ,
M Jq
where
(E32)
0)(E 33)
coshy(w) = cosh //'c o s h / / —s in h //' sinh //c o s
when n-> 00. This result is exact, the effect o f c disappearing by division w ith n and the
sum m ation converging to a definite integral.
In the special case o f quadratic symmetry, H = H \ and these results reduce to
coshy(<y) = co sh //co th /jT —cos <y,
(E34)
since it follows from (E 20) that
sinh //s in h H = cosh Z/tanh H = cosh Z/tanh
311(1
where
l n 2 c o sh H =
l n £ {1 + V ( 1 - * i sin2|i')?fll<i)’
kx — (2 sin h //)/c o sh 2/ / .
1
(E 35)
(E 36)
(E37)
W e cannot reduce (E36) any further, but the potential energy per molecule U can be
given in terms o f the tabulated functions. W e have
(E38)
which yields
c
sinh
358
W. K. BURTON AND OTHERS ON THE GROWTH OF CRYSTALS
From (E 36), (E37) and (E38) we finally obtain
S= 1
where
k \+ k 2 = 1,
+
c o th tf,
(E39)
K ^ K i k ^ = I"*” (1 —k\ sin2w) “<rfw,
(E40)
Jo
the latter integral being the complete elliptic integral o f the first kind; formula (E39) is
the formula (93) in the text.
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