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Computer simulation of spherical crystal surfaces: Defects in hexagonal lattices
SURFACE
SCIENCE 14 (1969) 423-430 0 North-Holland
Publishing Co., Amsterdam
COMPUTER SIMULATION OF SPHERICAL CRYSTAL SURFACES: DEFECTS IN HEXAGONAL LATTICES A. J. PERRY * Battelle Institute, Advanced Studies Center, Geneva, ~wit~er~a~ and D. G. BRANDON Department of Materials Engineering, The Technion, Haifa, lsrael Received 13 August 1968 Previous work on field-ion image simulation of lattice defects in cubic crystal+-5) using the thin shell mode16) has now been extended to the hexagonal case by substituting the atom positions appropriate to a hexagonal metal (ruthenium). By modifying the ideas of Melmed and Ranganathan7-8), the order of prominence of the lattice planes and the multiplicity of dislocation spirals can be accounted for in terms of the relative distances between A and B layers in the lattice. Satisfactory simulations has been achieved for dislocations, dislocation loops and dipoles, and partial dislocations and stacking faults. A program has also been written for twinned crystals. 1. Introduction
In previous workl-5) computer methods have been developed for simulating field-ion images of lattice defects in f.c.c. and b.c.c. metals. The programs so far written have covered unit lattice dislocations, dislocation loops and dipoles and stacking fault ribbons, and the present paper reports a further extension of these same programs and their application to the case of hexagonal metals. As in the authors’ previous calculations, the thin shell models) is used but now with the lattice positions and displacement vectors appropriate to the hexagonal unit cell. 2.
Method of
calcuiation
The previously described programs for the cubic system were adapted to the hexagonal case by writing the displacement vectors and the lattice positions for the hexagonal lattice in terms of orthogonal axes with the following orientation relation: co~~lo~~~~~~l~~ wiom~~~~H. * Present address: Brown Boveri Research Center, 5401 Baden, Switzerland. 423
A.J.PERRY
424
The Miller-Bravais the
3-index
identical W’=(c/a)
form:
vector
indices ti=l[-t,
in the
W, where
the
AND
of a vector V=D-t.
orthogonal unit
D.G.BRANDON
[u u t II.] were first converted
W=H’, system:
of distance
then
transformed
into
U’= - i U+ V. V’= -, in the
system is a. Similarly the normal to any plane whose (/? k il) was expressed in terms of the computer
orthogonal
to the j 17,
coordinate
Miller-Bravais indices are orthogonal cell by: h’=k,
I<’= - (2h + k)/,/3, I’ = (c/a, 1. Once the Miller-Bravais displacement vectors, lattice positions and lattice vectors had been converted into the orthogonal coordinate system, the atom displacement could be determined from the displacement equations given in previous publications. In all cases the value of c/u was taken as 1.582, appropriate to ruthenium. All calculations were programmed in Fortran for a CDC 6600 computer and the results were plotted, without rounding, on-line on a 565 Calcomp plotter. 3. Dislocations Melmed 7, and Ranganathan and Melmed s, have noted some of the peculiarities of the hexagonal lattice which arise from the two-valued interplanar spacing of many of the prominent lattice planes, and have proposed modifying the g -b criterion for predicting the multiplicity of the spiral generated by a dislocation in the hexagonal lattice. If the basal planes are labelled ABAB . . . . according to common practice, then for every lattice plane it is possible to determine the distance between neighbouring A planes, tiAA, in units of a, and to compare this with the shortest distance between an A and a B plane, dAAH.This has been done in table 1, using c/u= 1.582, and compared with the observed order of prominence of the lattice planes seen in published field-ion images of ruthenium7). The case dAjAB/dAA=Ocorresponds to a mixed AB plane, ciJdAAA=+ corresponds to equidistant A and B layers, while other values correspond to the rippled double layers discussed by Melmed’). It is these other values. dAAB/dAAA=+ or +, that correspond to two-valued interplanar spacings. It is not true to say that the simple g -b rule is invalid for the hexagonal lattice, rather the value of g must be correctly chosen so that lg/= I/C/~,. Then g .b will give the correct multiplicity for dAB/dAA=O and one half of the correct multiplicity for u’AB/d AA=+ (compare the order observed in table 1). However, if CjAB/dAA=+ or 4, then the effect is not just to double the multiplicity (as stated by Ranganathan and Melmeds)) but rather to generate sets of “tramlines”. The condition for visibility of the tramlines is determined by the shell thickness, and is simply the condition that succes-
425
sive A and B layers should not overlap in the thin shell. Separation of the tramlines cannot be observed if pOdAB, then the tramlines will be visible near the pole of the prominent planes, but will fade out at an angle 8 from the pole, corresponding to the point
at which p,, = dAB cos 8. TABLE 1
Order of importance (hk.1)
d&a
Oool lOT0 1011 loi
1.582 0.866 0.759 0.583 0.500 0.477 0.450 0.423 0.418 0.363 0.359 0.334 0.321 0.310 0.302
1120 1121 loi
1122 2021 1123 loi
2023 2151 1124 2132
of crystal planes d&d&t
Observed order 1 2 3 5 4 12 7 6 8 14 not observed 9 10 11 13
dnA values from 2/3 2~;
a
= 2/ [h2 + hk + k2 + 3 (a/c) PI,
using c/a = 1.582 ($(a/c)z = 0.300).
The slip dislocation commonly found in hexagonal crystals is 3 (1120) (type a) but two other unit lattice dislocations are also observed: (0001) (type c) and +( 1123) (type ~+a). Fig. 1 illustrates the variety of contrast that may be obtained: In fig. la a dislocation of Burgers’ vector +[2iiO] has been allowed to emerge on the (1120) plane to generate a single fold spiral (g - b = 1, dAB/dAA= =O). In fig. lb the same dislocation emerges on the (IOiO) plane (g -b= 1, dAB/dAA=+) and a pair of tramlines is visible over most of the single spiral. -In fig. lc a $[1213] dislocation is seen emerging on the (0001) plane (g-b= 1, dAB/dAA =+) and the doubling of the single spiral as a result of the intermediate B layer is seen. 4. Dislocation loops and dipoles The calculation
for loops
and
dipoles
is straightforward
and
follows
425
A.J.PEKKY
immediately
from the considerations
AND
D.G.
DKANDON
given above
for dislocations
and the
previous calculation for the cubic cases). Fig. 2a shows an example of a dipole centred on the (0001) planes with dislocation axes on the hypothetical (liO9)
poles.
The Burgers
:. . .. I.
vector
:’
**f’
.__
.
.
*
* .
:
I: . ; .. . . . : . .
1
. .
I
.
::
..
“,.
.I
.
-*
** *.
*.. : .
::.
*
.
:
.
.- .*. ** -.. --. **a ...* .. **. * ::* . . . a.. . . .. . ‘.. .
*.
. .; *:, a... ‘+ .. ‘*;* *.. .
:
and this
*
::
.. ..
1, ~/~a/~~~=$)
*--. .*.** _* .*
::
is [OOOl] (g-b=
I
.
. .*
*.*
. . . *: .. . * *
. .
.
-.
:.
.
:* :
:
-
-
-.
Fig. 1. Dislocation of Burger’s vector 4 [ZfiO] emerging on (a) the (I 120) plane (~-6 1, c/~~~]~.~A -- 0) and (b) the (lOi0) plane (S./I -- I, d.ttl/d,t,, -f).(c) A :,[f2i31 dislocation emerging on the (0001) plane (g.15 I, n’,\lr/dlt-\ 4). Tip radius 143~ in all cases.
case obeys the rule given by Ranganathan and Melmeds). In fig. 2b a slightly more complicated case is shown: the Burgers vector is +[1210] and the exit pole (i103) (g*b= I, rt;z,/~AA=&). In this case the tramlines are invisible and the (il03) planes are rippled and not separated into A and B layers (y. >dAa and the layers overlap in the image), In practice it is often more convenient to treat dislocation loops and dipoles as a special case of a multiple dislocation array. The computed
COMPUTER
centre of the simulated of the constituent
SIMULATION
OF SPHERICAL
CRYSTAL
image is then chosen independently
dislocations.
The displacements
421
SURFACES
of the exit poles
due to each dislocation
are then calculated separately and summed to yield the final atom positions in the dislocated crystal. This technique has been used to simulate actual arrays in f.c.c. crystals5).
Fig. 2. Dislocation dipoles: (a) Burgers’ vector [OOOl], emerging on (0001) planes. (g-b = 1, d_&dAA =+, tip radius 804. (b) Burgers’ vector +[T2iO], emerging on (1103) (g-b = 1, dAB/dAA= 4, tip radius 150~1).
5. Stacking faults Stacking faults can form on the basal plane of hexagonal crystals as the result of the dissociation of a glissile dislocation (+[i2iO]+$[OliO]+ ++[ilOO]) or by point defect condensation. Only the former case has been programmed, since the case of point defect condensation is effectively included in the program for loops and dipoles and is obtained by inserting the appropriate partial Burgers vector in the read-in. As in f.c.c. crystals, some care is needed in the identification of stacking fault ribbons because of the sensitivity of the contrast to the exact position and orientation of the line of intersection of the fault with the surface. Fig. 3 illustrates a case in point: a dislocation of Burgers vector 3 [i2iO] has dissociated into the two partials given above, b, =$z[01iO], b,=+a[ilOO] with a separation of 4~. The fault emerges on the (2114) plane, so that g - b, = 0, g ~6, = 1. The contrast from the undissociated dislocation, fig. 3a, is very little different from that from the dissociated ribbon, fig. 3b.
6. Twinning Twinning
is a major
deformation
mode
in hexagonal
crystals
and the
A. J. PERRY
428
identification
of twinned
materials. A program been written. In principle
structures
AND D. G. BRANDON
is therefore
for the simulation
it is possible
to simulate
of some importance
of twinned any boundary
crystals
in these
has therefore
by selecting
the centre
Fig. 3. Dissociation into a stacking fault ribbon: (a) Perfect dislocation of Burgers’ - 0). (b) The same disvector f-[1210] emerging on the (21 14) plane (g.h = 1, dl~~~/d.za location after dissociation into two partials on the basal plane. 3 [i2iO]~ ’ .\ [Ol TO] ‘m -I~+[ilOO] (g.bl0, g-62 I). Tip radius 150~1.
of the image for one grain, transforming to this axis in the second grain, and then running both cases and using a pair of scissors on the results. In actual fact twinning in the hexagonal lattice cannot readily be handled in this way. The basic problem is that the twin cannot be formed by simple shear - the atoms have to rearrange (“shuffle”). It follows that a twinning dislocation with a Burgers vector as defined by the twinned structure will not in fact translate the atoms into the final twinned positions. A secondary problem, not usually considered, is that a lattice twin and a crystallographic twin are not necessarily the same thing. A crystallographic twin is formed when crystal planes and directions in the two grains are twin-related, but a lattice twin requires that each atom in the matrix lattice should find a corresponding atom position in the twinned lattice. In hexagonal crystals this is almost certainly not the case, the minimum twin boundary energy corresponding instead to a matrix lattice, ABAB . . . . being “twinned” into the lattice positions, BCBC . . . or CACA . . I. In other words the twinned lattice may be displaced bodily by the lattice vector +( IiOO). There are three possible ways of simulating the twinned structure: a) shear and shuffle; b) a rotation based on the coincidence or O-lattice transformationg);
COllPPUTER
SI~LATION
OF SPHERlCAL
CRYSTAL
SURFACES
429
c) a mirror reflection in the composition plane, plus a lattice displacement if necessary. The last is the easiest to program, but has the disadvantage that effects associated with the growth of the twin cannot be simulated. That is, the twin boundary is constrained to be coherent and twinning dislocations are not present. A particular composition plane defined by the direction cosines lmn in the computer orthogonal axes, is constrained to pass through the pole on the surface with direction cosines pqr in a crystal of radius R. An atom at t is then reflected to the twinned lattice position ,F, given by F==t+2[R(Ep+mq+nr
-t*n]n,
where n is the normalized vector [pqr]. The thin shell test is then performed on each reflected atom and the results centred on any convenient pole. The extension to a displaced lattice only requires substituting F’= F-4- 6, where b is the lattice displacement vector. For c/a< I .633, (lOi2) and (11%) are possible composition planes. Fig. 4a
Fig. 4. Coherent twin boundaries passing through the (i01.5) plane. (a) (1012) composition plane, twinning direction [Toll]; (b) (1122) composition plane, twinning direction [TT23].
shows (lOi2) twin boundary emerging on the (iOl5) plane while fig. 4b shows a (I 122) twin boundary emerging on the same plane. In fig. 4a the twinning direction is tangential to the exit plane. 7. Discussion There appears to be no limit to the possibilities of computer simulation for field-ion images of lattice defects although the application of these
430
A. J. PERRY
techniques
to actual
AND
experimental
D. G. BRANDON
results
is not so simple.
The extension
of previous work on cubic metals to the case of the hexagonal lattice has revealed some interesting features but is in the main straightforward. Most of the peculiarities
of the hexagonal
lattice are connected
with the spacing
of alternate A and B layers, with dAB/d,4A=0, $, $: or -f depending on g. For dAB/dA,#O the multiplicity of dislocation spirals may be doubled or tramlines may appear. The condition for invisibility of the tramlines is just dA,
8. Conclusions I) Dislocations, dislocation loops and dipoles, stacking fault ribbons and coherent twin boundaries have been successfuhy simulated in the hexagonal close packed lattice. 2) The effect of the hexagonal close packing on the g 6 criterion for dislocation spirals has been reconsidered and the conclusions of Ranganathan and Melmeds) modified accordingly. 3) The conditions for visibility of the above defects have been discussed within the framework of the thin shell model for defect lattices. l
References 1) 2) 3) 4) 5) 6) 7) 8) 9)
D. G. Brandon and A. J. Perry, Phil. Mag. 16 (1967) 131. A. J. Perry and D. G. Brandon, Phil. Mag. 17 (1968) 255. A. J. Perry and D. G. Brandon, Phil. Mag. (1968) to be published. R. C. Sanwald, S. Ranganathan and J. J. Hren, Appl. Phys. Letters 9 (1966) 393. R. C. Sanwald and J. J. Hren, Surface Sci. 9 (1968) 257. A. J. W. Moore, J. Phys. Chem. Solids 23 (1962) 907. A. J. Melmed, Surface Sci. 5 (1966) 359. S. Ranganathan and A. J. Melmed, Phil. Mag. 15 (1966) 1309. W. Bollman, Phil. Mag. 16 (1967) 363, 383.
SCIENCE 14 (1969) 423-430 0 North-Holland
Publishing Co., Amsterdam
COMPUTER SIMULATION OF SPHERICAL CRYSTAL SURFACES: DEFECTS IN HEXAGONAL LATTICES A. J. PERRY * Battelle Institute, Advanced Studies Center, Geneva, ~wit~er~a~ and D. G. BRANDON Department of Materials Engineering, The Technion, Haifa, lsrael Received 13 August 1968 Previous work on field-ion image simulation of lattice defects in cubic crystal+-5) using the thin shell mode16) has now been extended to the hexagonal case by substituting the atom positions appropriate to a hexagonal metal (ruthenium). By modifying the ideas of Melmed and Ranganathan7-8), the order of prominence of the lattice planes and the multiplicity of dislocation spirals can be accounted for in terms of the relative distances between A and B layers in the lattice. Satisfactory simulations has been achieved for dislocations, dislocation loops and dipoles, and partial dislocations and stacking faults. A program has also been written for twinned crystals. 1. Introduction
In previous workl-5) computer methods have been developed for simulating field-ion images of lattice defects in f.c.c. and b.c.c. metals. The programs so far written have covered unit lattice dislocations, dislocation loops and dipoles and stacking fault ribbons, and the present paper reports a further extension of these same programs and their application to the case of hexagonal metals. As in the authors’ previous calculations, the thin shell models) is used but now with the lattice positions and displacement vectors appropriate to the hexagonal unit cell. 2.
Method of
calcuiation
The previously described programs for the cubic system were adapted to the hexagonal case by writing the displacement vectors and the lattice positions for the hexagonal lattice in terms of orthogonal axes with the following orientation relation: co~~lo~~~~~~l~~ wiom~~~~H. * Present address: Brown Boveri Research Center, 5401 Baden, Switzerland. 423
A.J.PERRY
424
The Miller-Bravais the
3-index
identical W’=(c/a)
form:
vector
indices ti=l[-t,
in the
W, where
the
AND
of a vector V=D-t.
orthogonal unit
D.G.BRANDON
[u u t II.] were first converted
W=H’, system:
of distance
then
transformed
into
U’= - i U+ V. V’= -, in the
system is a. Similarly the normal to any plane whose (/? k il) was expressed in terms of the computer
orthogonal
to the j 17,
coordinate
Miller-Bravais indices are orthogonal cell by: h’=k,
I<’= - (2h + k)/,/3, I’ = (c/a, 1. Once the Miller-Bravais displacement vectors, lattice positions and lattice vectors had been converted into the orthogonal coordinate system, the atom displacement could be determined from the displacement equations given in previous publications. In all cases the value of c/u was taken as 1.582, appropriate to ruthenium. All calculations were programmed in Fortran for a CDC 6600 computer and the results were plotted, without rounding, on-line on a 565 Calcomp plotter. 3. Dislocations Melmed 7, and Ranganathan and Melmed s, have noted some of the peculiarities of the hexagonal lattice which arise from the two-valued interplanar spacing of many of the prominent lattice planes, and have proposed modifying the g -b criterion for predicting the multiplicity of the spiral generated by a dislocation in the hexagonal lattice. If the basal planes are labelled ABAB . . . . according to common practice, then for every lattice plane it is possible to determine the distance between neighbouring A planes, tiAA, in units of a, and to compare this with the shortest distance between an A and a B plane, dAAH.This has been done in table 1, using c/u= 1.582, and compared with the observed order of prominence of the lattice planes seen in published field-ion images of ruthenium7). The case dAjAB/dAA=Ocorresponds to a mixed AB plane, ciJdAAA=+ corresponds to equidistant A and B layers, while other values correspond to the rippled double layers discussed by Melmed’). It is these other values. dAAB/dAAA=+ or +, that correspond to two-valued interplanar spacings. It is not true to say that the simple g -b rule is invalid for the hexagonal lattice, rather the value of g must be correctly chosen so that lg/= I/C/~,. Then g .b will give the correct multiplicity for dAB/dAA=O and one half of the correct multiplicity for u’AB/d AA=+ (compare the order observed in table 1). However, if CjAB/dAA=+ or 4, then the effect is not just to double the multiplicity (as stated by Ranganathan and Melmeds)) but rather to generate sets of “tramlines”. The condition for visibility of the tramlines is determined by the shell thickness, and is simply the condition that succes-
425
sive A and B layers should not overlap in the thin shell. Separation of the tramlines cannot be observed if pO
at which p,, = dAB cos 8. TABLE 1
Order of importance (hk.1)
d&a
Oool lOT0 1011 loi
1.582 0.866 0.759 0.583 0.500 0.477 0.450 0.423 0.418 0.363 0.359 0.334 0.321 0.310 0.302
1120 1121 loi
1122 2021 1123 loi
2023 2151 1124 2132
of crystal planes d&d&t
Observed order 1 2 3 5 4 12 7 6 8 14 not observed 9 10 11 13
dnA values from 2/3 2~;
a
= 2/ [h2 + hk + k2 + 3 (a/c) PI,
using c/a = 1.582 ($(a/c)z = 0.300).
The slip dislocation commonly found in hexagonal crystals is 3 (1120) (type a) but two other unit lattice dislocations are also observed: (0001) (type c) and +( 1123) (type ~+a). Fig. 1 illustrates the variety of contrast that may be obtained: In fig. la a dislocation of Burgers’ vector +[2iiO] has been allowed to emerge on the (1120) plane to generate a single fold spiral (g - b = 1, dAB/dAA= =O). In fig. lb the same dislocation emerges on the (IOiO) plane (g -b= 1, dAB/dAA=+) and a pair of tramlines is visible over most of the single spiral. -In fig. lc a $[1213] dislocation is seen emerging on the (0001) plane (g-b= 1, dAB/dAA =+) and the doubling of the single spiral as a result of the intermediate B layer is seen. 4. Dislocation loops and dipoles The calculation
for loops
and
dipoles
is straightforward
and
follows
425
A.J.PEKKY
immediately
from the considerations
AND
D.G.
DKANDON
given above
for dislocations
and the
previous calculation for the cubic cases). Fig. 2a shows an example of a dipole centred on the (0001) planes with dislocation axes on the hypothetical (liO9)
poles.
The Burgers
:. . .. I.
vector
:’
**f’
.__
.
.
*
* .
:
I: . ; .. . . . : . .
1
. .
I
.
::
..
“,.
.I
.
-*
** *.
*.. : .
::.
*
.
:
.
.- .*. ** -.. --. **a ...* .. **. * ::* . . . a.. . . .. . ‘.. .
*.
. .; *:, a... ‘+ .. ‘*;* *.. .
:
and this
*
::
.. ..
1, ~/~a/~~~=$)
*--. .*.** _* .*
::
is [OOOl] (g-b=
I
.
. .*
*.*
. . . *: .. . * *
. .
.
-.
:.
.
:* :
:
-
-
-.
Fig. 1. Dislocation of Burger’s vector 4 [ZfiO] emerging on (a) the (I 120) plane (~-6 1, c/~~~]~.~A -- 0) and (b) the (lOi0) plane (S./I -- I, d.ttl/d,t,, -f).(c) A :,[f2i31 dislocation emerging on the (0001) plane (g.15 I, n’,\lr/dlt-\ 4). Tip radius 143~ in all cases.
case obeys the rule given by Ranganathan and Melmeds). In fig. 2b a slightly more complicated case is shown: the Burgers vector is +[1210] and the exit pole (i103) (g*b= I, rt;z,/~AA=&). In this case the tramlines are invisible and the (il03) planes are rippled and not separated into A and B layers (y. >dAa and the layers overlap in the image), In practice it is often more convenient to treat dislocation loops and dipoles as a special case of a multiple dislocation array. The computed
COMPUTER
centre of the simulated of the constituent
SIMULATION
OF SPHERICAL
CRYSTAL
image is then chosen independently
dislocations.
The displacements
421
SURFACES
of the exit poles
due to each dislocation
are then calculated separately and summed to yield the final atom positions in the dislocated crystal. This technique has been used to simulate actual arrays in f.c.c. crystals5).
Fig. 2. Dislocation dipoles: (a) Burgers’ vector [OOOl], emerging on (0001) planes. (g-b = 1, d_&dAA =+, tip radius 804. (b) Burgers’ vector +[T2iO], emerging on (1103) (g-b = 1, dAB/dAA= 4, tip radius 150~1).
5. Stacking faults Stacking faults can form on the basal plane of hexagonal crystals as the result of the dissociation of a glissile dislocation (+[i2iO]+$[OliO]+ ++[ilOO]) or by point defect condensation. Only the former case has been programmed, since the case of point defect condensation is effectively included in the program for loops and dipoles and is obtained by inserting the appropriate partial Burgers vector in the read-in. As in f.c.c. crystals, some care is needed in the identification of stacking fault ribbons because of the sensitivity of the contrast to the exact position and orientation of the line of intersection of the fault with the surface. Fig. 3 illustrates a case in point: a dislocation of Burgers vector 3 [i2iO] has dissociated into the two partials given above, b, =$z[01iO], b,=+a[ilOO] with a separation of 4~. The fault emerges on the (2114) plane, so that g - b, = 0, g ~6, = 1. The contrast from the undissociated dislocation, fig. 3a, is very little different from that from the dissociated ribbon, fig. 3b.
6. Twinning Twinning
is a major
deformation
mode
in hexagonal
crystals
and the
A. J. PERRY
428
identification
of twinned
materials. A program been written. In principle
structures
AND D. G. BRANDON
is therefore
for the simulation
it is possible
to simulate
of some importance
of twinned any boundary
crystals
in these
has therefore
by selecting
the centre
Fig. 3. Dissociation into a stacking fault ribbon: (a) Perfect dislocation of Burgers’ - 0). (b) The same disvector f-[1210] emerging on the (21 14) plane (g.h = 1, dl~~~/d.za location after dissociation into two partials on the basal plane. 3 [i2iO]~ ’ .\ [Ol TO] ‘m -I~+[ilOO] (g.bl0, g-62 I). Tip radius 150~1.
of the image for one grain, transforming to this axis in the second grain, and then running both cases and using a pair of scissors on the results. In actual fact twinning in the hexagonal lattice cannot readily be handled in this way. The basic problem is that the twin cannot be formed by simple shear - the atoms have to rearrange (“shuffle”). It follows that a twinning dislocation with a Burgers vector as defined by the twinned structure will not in fact translate the atoms into the final twinned positions. A secondary problem, not usually considered, is that a lattice twin and a crystallographic twin are not necessarily the same thing. A crystallographic twin is formed when crystal planes and directions in the two grains are twin-related, but a lattice twin requires that each atom in the matrix lattice should find a corresponding atom position in the twinned lattice. In hexagonal crystals this is almost certainly not the case, the minimum twin boundary energy corresponding instead to a matrix lattice, ABAB . . . . being “twinned” into the lattice positions, BCBC . . . or CACA . . I. In other words the twinned lattice may be displaced bodily by the lattice vector +( IiOO). There are three possible ways of simulating the twinned structure: a) shear and shuffle; b) a rotation based on the coincidence or O-lattice transformationg);
COllPPUTER
SI~LATION
OF SPHERlCAL
CRYSTAL
SURFACES
429
c) a mirror reflection in the composition plane, plus a lattice displacement if necessary. The last is the easiest to program, but has the disadvantage that effects associated with the growth of the twin cannot be simulated. That is, the twin boundary is constrained to be coherent and twinning dislocations are not present. A particular composition plane defined by the direction cosines lmn in the computer orthogonal axes, is constrained to pass through the pole on the surface with direction cosines pqr in a crystal of radius R. An atom at t is then reflected to the twinned lattice position ,F, given by F==t+2[R(Ep+mq+nr
-t*n]n,
where n is the normalized vector [pqr]. The thin shell test is then performed on each reflected atom and the results centred on any convenient pole. The extension to a displaced lattice only requires substituting F’= F-4- 6, where b is the lattice displacement vector. For c/a< I .633, (lOi2) and (11%) are possible composition planes. Fig. 4a
Fig. 4. Coherent twin boundaries passing through the (i01.5) plane. (a) (1012) composition plane, twinning direction [Toll]; (b) (1122) composition plane, twinning direction [TT23].
shows (lOi2) twin boundary emerging on the (iOl5) plane while fig. 4b shows a (I 122) twin boundary emerging on the same plane. In fig. 4a the twinning direction is tangential to the exit plane. 7. Discussion There appears to be no limit to the possibilities of computer simulation for field-ion images of lattice defects although the application of these
430
A. J. PERRY
techniques
to actual
AND
experimental
D. G. BRANDON
results
is not so simple.
The extension
of previous work on cubic metals to the case of the hexagonal lattice has revealed some interesting features but is in the main straightforward. Most of the peculiarities
of the hexagonal
lattice are connected
with the spacing
of alternate A and B layers, with dAB/d,4A=0, $, $: or -f depending on g. For dAB/dA,#O the multiplicity of dislocation spirals may be doubled or tramlines may appear. The condition for invisibility of the tramlines is just dA,
8. Conclusions I) Dislocations, dislocation loops and dipoles, stacking fault ribbons and coherent twin boundaries have been successfuhy simulated in the hexagonal close packed lattice. 2) The effect of the hexagonal close packing on the g 6 criterion for dislocation spirals has been reconsidered and the conclusions of Ranganathan and Melmeds) modified accordingly. 3) The conditions for visibility of the above defects have been discussed within the framework of the thin shell model for defect lattices. l
References 1) 2) 3) 4) 5) 6) 7) 8) 9)
D. G. Brandon and A. J. Perry, Phil. Mag. 16 (1967) 131. A. J. Perry and D. G. Brandon, Phil. Mag. 17 (1968) 255. A. J. Perry and D. G. Brandon, Phil. Mag. (1968) to be published. R. C. Sanwald, S. Ranganathan and J. J. Hren, Appl. Phys. Letters 9 (1966) 393. R. C. Sanwald and J. J. Hren, Surface Sci. 9 (1968) 257. A. J. W. Moore, J. Phys. Chem. Solids 23 (1962) 907. A. J. Melmed, Surface Sci. 5 (1966) 359. S. Ranganathan and A. J. Melmed, Phil. Mag. 15 (1966) 1309. W. Bollman, Phil. Mag. 16 (1967) 363, 383.