ultramicroscopy ELSEVIER
Ultramicroscopy 56 (1994) 216-224
In-situ observations of irradiation-induced phase transformations C. K i n o s h i t a a, y . T o m o k i y o
b K.
Nakai c
Department of Nuclear Engineering, Faculty of Engineering, Kyushu University 36, Fukuoka 812, Japan b Research Laboratory of High Voltage Electron Microscopy, Kyushu University 36, Fukuoka 812, Japan c Department of Materials Science and Engineering, Ehime University, Matsuyama 790, Japan Received in final form at the Editorial Office 30 June 1994
Abstract High-voltage and high-resolution electron microscopy approaches the resolving power of 0.1 nm and is expected to develop new fields of materials science and engineering. In order to look over future advances, the present paper introduces the progress of our studies on irradiation-induced phase transformations in alloys based on conventional high-voltage electron microscopy. This paper also shows phase transformations at or near surface of a-A1203 induced by an intense electron beam in clean high-vacuum microscopes which give insight into irradiation-induced phenomena in new high-voltage and high-resolution electron microscopy.
1. Introduction Electron microscopy provides structural information and chemical analysis of materials through elastic and inelastic interactions between electrons and materials [1-3]. The same scattering process induces radiation effects in materials. Electron microscopy, in particular high-voltage electron microscopy (HVEM), gives dynamic and quantitative information about radiation effects through easy control of a wide variety of experimental parameters, such as electron energy, electron flux, irradiation time, irradiation temperature, crystallographic orientation and irradiation position [4]. The radiation effects depend strongly on atmospheric conditions, and their consequences in ultra-vacuum electron microscopes might be different from those in conventional electron microscopes, especially at and near specimen surfaces [5,6]. The objective of the present paper is to review our recent progress on irradiation-induced phase
transformations through free point defects and on those at a n d / o r near specimen surfaces.
2. Irradiation-induced phase through free point defects
transformations
In atomic displacement processes, replacement collision sequences play an important role as a means for transporting energy and mass through crystalline materials. Most of the displacements occur via replacement collision sequences, leaving a vacancy at the beginning of the sequence and depositing the excess atom as an interstitial at the end of the sequence. Electron irradiation induces rather isolated Frenkel pairs of interstitials and vacancies as well as electronic excitation. Most of these point defects, however, have a short lifetime and annihilate via interstitial-vacancy recombination. When superlattice alloys such as
0304-3991/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 3 9 9 1 ( 9 4 ) 0 0 1 0 2 - 2
C. Kinoshita et al. / Ultramicroscopy 56 (1994) 216-224
FeAI and Ni3A1 are irradiated with high-energy electrons at low temperatures, where interstitials and vacancies are not sufficiently mobile, the displacement a n d / o r replacement collisions are accompanied by chemical disordering. Two chemical disordering mechanisms are involved: (1) replacement collisions along mixed atom rows; and (2) the recombination of displaced atoms with vacancies on the "wrong" sublattice. Mechanisms (1) and (2) have been confirmed to be dominant in L12-type superlattice [7] and in B2-type superlattice having structural vacancies due to nonstoichiometry [8], respectively. The effects of electron irradiation on phase stability have been extensively studied in alloys, not only because of scientific interest, but also for the assessement of nuclear materials. Two types of irradiation-induced precipitation have been found in many systems, and several mechanisms have been suggested to explain these phenomena. One is inhomogeneous precipitation around point defect sinks. This type irradiation-induced precipitation has been succesfully interpreted as resulting from the accumulation of solute atoms that
217
have drifted toward defect sinks or the matrix. The other is homogeneous precipitation in the form of coherent or incoherent precipitates. Electron irradiation also induces modulated structures in a - F e - M o , A u - N i , C u - N i and C u Ti alloys even at temperatures above the spinodal temperature of each alloy [9]. In this section, irradiation-induced spinodal decomposition is introduced, following experimental results mainly on A u - N i alloys. Spinodal decomposition is expected during annealing only below the coherent spinodal curve after quenching from solutiontreatment temperatures. A u - N i alloys were irradiated with 1000 keV electrons at temperatures below and above the chemical spinodal temperatures. Fig. 1 is a typical dark-field micrograph showing the modulated structure along the (100) direction and satellite spots along the same direction on the corresponding diffraction pattern. The wavelength of the modulated structure )t remains constant for increasing irradiation time after rapid formation of the modulated structure [9]. Such kinetic behavior, according to the Cahn-Hilliard theory [10], suggests a modulated structure being
Fig. 1. Dark-field electron micrograph and the corresponding diffraction pattern showing modulated structure in A u - 7 0 a t % N i .
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induced by spinodal decomposition. The C a h n Hilliard theory gives the wave number of the modulated structure /3m (= 2~-/;t) in terms of the spinnodal temperature T~ as /32 o~ ( T - Ts),
(1)
where T is the annealing temperature. If the equation holds for spinodal decomposition under irradiation, a plot of observed/32 versus T, where T is the irradiation temperature in this case, should be linear and should intersect the T-axis at Ts. The value of Ts under irradiation, which is defined as Tsirr, increases with increasing electron flux 4, [9]. The maximum spinodal temperature with respect to 4' reaches up to the chemical spinodal temperature. It is well known that coherent spinodal decomposition maintains coherency under composition modulation and induces expansive and compressive strain fields periodically at the interval of compositional wave length. Such kind of modulation leads interstitials and vacancies, respectively, into expansive and compressive strain fields to
reduce the strain energy under irradiation. The full relaxation of the coherent strain increases the spinodal temperature by the difference between the chemical and the coherent spinodal temperatures (T5c and T~*) and is related by 2 Tl2Y(hkl)
(T~C-Ts*)
S"
'
(2)
where -,7 and S" are, respectively, the partial derivative of lattice constant and the second derivative of entropy with respect to composition, and Y is a function of the elastic constant [11]. The results of increment of the spinodal temperature due to 1000 keV electron irradiation, a(Ts*)irr, are in good agreement with calculated values of (T~C-T~*) within experimental errors [9]. This tends to confirm that the relaxation of the coherent strain associated with the modulated structure expands the spinodal region, and that electron irradiation relaxes the coherent strain. It is then instructive to examine the variation of lattice sites in the modulated structure caused
v
Fig. 2. H R E M image taken with four 200 reflections, their satellite spots and the transmitted beam in an Au-70at%Ni alloy having a modulated structure.
C. Kinoshita et al. / Ultramicroscopy 56 (1994) 216-224
by irradiation. The information on the lattice strain allows an investigation of the behavior of point defects. Lattice arrangements of the modulated structure during irradiation are studied through high-resolution electron microscopy. A t o m a n d / o r lattice displacements are expected by relaxation of the coherent strain under irradiation. A JEM-4000EX was used for taking H R E M images from an identical area of the modulated structure in A u - 7 0 a t % N i during 400 keV electron irradiation at room temperature; one example is shown in Fig. 2. The transmitted beam and four {200} reflections with their satellites were used for images under the conditions: spherical aberration constant of the objective lens C~ = 1 ram, divergence of the b e a m a = 3 x 10 - 4 rad, underfocus /xf = - 5 0 nm and specimen thickness t --- 15 nm. U n d e r 400 keV electron irradiation, the production rate of point defects is small enough that the wavelength, 2t0, of the original modulated structure hardly varies. On the other hand, the fluctuation of (200) interplanar spacing against distance clearly diminishes with increasing irradiation time, resulting in an interplanar spacing getting close to be contrast, as shown in Fig. 3. The striations corresponding to the composition modulation still remain even after disappearance of the phase modulation. The result suggests that the modulation of the lattice sites reduces in spite of the existence of composition modulation under irradiation. The relaxation of the coherent strain associated with the modulated structure occurs without destruction of coherency between lattice planes through the alternate accumulation of interstitials and vacancies with a periodicity of 3,0 into expansive and compressive regions, respectively. The C I H R T E M code [12] was used with experimental conditions such as C s, a, A f and t for the simulation of the H R E M image projected along the [001] direction of A u - 3 0 a t % N i on the basis of the following models for the atomic arrangement: (1) one-dimensional composition modulation along the [010] direction with an amplitude of 20at%Au and a wavelength of 1.87 nm, which is ten times the (020) interplanar spacing, (2) two-dimensional composition modulation along the [010] and the [001] directions having the
219
0.22
"~ 0.16 0.14
I
E=400keV
Ni 30 O a t * / Au
ko = 1 . 7 x 1 0 2 3 e / m 2 s
(D 0.18 v
c~ 016
c "G
tirr= 2220S (B)
014 ol
o.2oF { 038 0.16 ~| 0.14~ 0
{
÷.+÷
2 ,J 10
4 ,' 20
tirr=3180s 6 nm (C) , ' , 30 40 50
Distance (Number of (200) Lattice planes )
Fig. 3. Variation of (200) interplanar spacing with distance along the [100] direction under 400 keV electron irradiation at 298 K in A u - 7 0 a t % N i alloy. (A), (B) and (C) were taken, respectively, at tir r = 1140, 2220 and 3180 s. The interplanar spacing was determined using the H R E M image taken from the identical area under irradiation. The value of A0 corresponds to the wavelength of the modulated structure existing before irradiation.
same amplitude and wavelength as those in (1), and (3) two-dimensional phase modulation along the [010] and the [001] directions under A L / L = 0.03 without composition modulation, where AL is the maximum shift of atom positions from the regular site and L the (020) interplanar spacing without modulation. White dots appear between (020) planes. The spacing between white dots for models (2) and (3) is respectively plotted in Figs. 4 and 5. In the case of model (2), the dot spacing stays constant (Fig. 4), in spite of the compositional modulation. The simulation was not performed on a three-dimensional model, but it should be emphasized that the modulation of dot-contrast modulation on the structure image is mainly due to the phase modulation as shown in model (3).
C. Kinoshita et al. / Ultramicroscopy 56 (1994) 216-224
220 § 1.25
# 125
Two-dimensional
-e
phase- modulation
Two-dimensionat composition-modulation
cm
1.00
IOC
2 e
0.750
I
5 Position (the number of interval b e t w e e n
1
I0 white dots)
Fig. 4. Variation of the (020) interplanar spacing, d, with position, simulated based on model (2), where d020 is the (020) interplanar spacing without modulation.
In order to confirm the periodical distribution of point defects in the modulated structure, nucleation sites of loops were examined through weak-beam electron microscopy. Irradiation with 1000 keV electrons was performed at room temperature on an A u - 7 0 a t % N i alloy after inducing the modulation under irradiation at 690 K, that is, above the coherent spinodal t e m p e r a t u r e without irradiation, Ts. The nucleation of tiny interstitial loops of less than 2 nm diameter occurs in rows at a regular interval as shown in Fig. 6. The
~ 0,7=.
I I 5 I0 Position (the number of interval between white dots)
Fig_ 5. Same as in Fig_ 4, but based on model (3).
interval between the rows coincides with the wavelength recognized in the induced modulated structure at 690 K. The periodic nucleation of interstitial loops reflects the periodic distribution of interstitials, and supports the relaxation mechanism of the coherent strain.
3. Irradiation-induced phase transformation in high-vacuum TEM Irradiation-induced phase transformations are very sensitive to the atmospheric conditions in
Fig. 6_ Weak-beam electron micrograph taken from an A u - 7 0 a t % N i alloy, which was irradiated with 1000 keV electrons at room temperature after inducement of modulated structure at 690 K. Dot contrasts correspond to small interstitial loops.
C. Kinoshita et al. / Ultramicroscopy 56 (1994) 216-224
221
Fig. 7. Typical example of HREM images of a-AI20 3 along the [110] direction and the corresponding diffraction pattern, showing line contrasts parallel to the (001) basal plane.
electron microscopes. Oxidation and reduction are frequently observed in conventional electron microscopes. Clean high-vacuum also provides new types of phase transformations under electron irradiation. An example is a structural change induced by electron irradiation at or near the surface of a - A l 2 0 3 specimens in clean-vacuum electron microscopes [5,6]. When a - A I 2 0 3 thin foils are observed in a JEM-4000 EX, mottled contrast appears within several minutes. The contrast becomes wavy, or like a "patchwork quilt" [6,13] with an increase in
observation time. Then holes and dark-line contrast appear. Anisotropic hole drilling was observed in MgO even when circular electron probes were employed [14]. The mottled or patchworkquilt-like contrast can be seen at 200 keV too, suggesting that the roughening and hole drilling originate from desorption induced by electronic transitions at surfaces. During observation along the [110] direction at 400 keV, dark-line contrast appears as shown in Fig. 7. The number density of dark-lines increases with increasing irradiation time. The dark-line contrast appears in a thicker
Fig. 8. HREM image of a-Al20 3 and the corresponding electron diffraction pattern taken along the [001] direction at 400 keV_ Note periodic modulation of bright dots in intensity especially inside the circled areas and extra spots in the diffraction pattern.
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C. Kinoshita et al. / Ultrarnicroscopy 56 (1994) 216-224
region as well as a thin area, though the contrast is not always dark. The lines are parallel to the (001) basal plane, showing streaks along the [001] direction in the corresponding diffraction pattern. The streaks are observable when many darkor bright-lines appear and suggest the presence of planar defects [5]. These dark- or bright-lines were also produced at 200 keV. Bursill et al. [13] observed similar results even at 100 keV which is much lower than the displacement energy for aluminum or oxygen ions. On the other hand, no line contrast appears even after 3600 s of electron irradiation at 200 keV in a JEM-2000FX, where the vacuum around the specimen is estimated to be about one order magnitude worse than in the JEM-4000EX. These results indicate that the formation of planar defects is very sensitive to the quality of vacuum around the specimen. In order to investigate the origin of the line contrast, H R E M images were observed along the [001] direction. Fig. 8 shows an example obtained at 400 keV. One can see no clear contrast which may correspond to edges of the dark- or brightlines. A periodic modulation in the intensity of bright dots can, however, be seen along three directions. Brighter dots appear every other lattice plane along the (110), (120) or (210) plane in
a
~"
b
e o
e
o
e
006
o
e
030
o
o
0
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o
o
0
O o
0 e
o
O
o o
•
0~
e
e 110 o
o
~
o
o
o 0
o
o
o
O
~T0 o
o
o
o0o
e 0
• o
e
e o
O
0 o
e
300 o
o o 2T0 o o o
T~0 o
•
o
e
3~o •
Fig. 9_ Key diagrams of the diffraction patterns for the [110] zone axis (a) and the [001] zone axis (b), The closed and open circles correspond to the normal and extra spots, respectively.
some regions. The configuration of brighter dots has sixfold symmetry. This periodic modulation can not be seen at an initial stage of electron irradiation but appears with increasing irradiation time. The size of the modulated region is comparable with the length of the dark- or bright-line. Extra spots are evident in the diffraction pattern shown in Fig. 8. They appear at ½(110), ½(300) and the identical position, indicating the presence of an ordered structure with a double period. These extra spots are also reproduced well by optical diffraction of the image of Fig. 8. A key for the diffraction pattern is shown in Fig. 9 along with the one for [110] diffraction pattern. The extra spots cannot be seen at the initial stage, and the intensity of the spots becomes strong with increasing irradiation time. It is clear therefore that the line contrast in Fig. 7 and the periodic modulation in Fig. 8 are due to planar defects having a double period ordered arrangement of atoms in a plane parallel to the (001) basal plane. It is worthwhile to recall the structure of perfect a - A l 2 0 3 before interpreting the contrast previously shown, a-A120 3 is well known to have a corundum structure (a 0 = 0.4758, c 0 = 1.2991 nm) in which O ions have a close-packed hexagonal arrangement, and AI ions occupy two third of the octahedral interstices formed by the O ions. One third of the octahedral interstices are vacant (the vacand site is referred to as an A1 vacancy). The sequence of layers of O ions is expressed is expressed as ABABAB..., while the one of AI ions as a / 3 7 a / 3 7 . . . A1 ions are actually displaced from the ideal positions along the c-axis and O ions are displaced within the (001) plane because of the presence of A1 vacancies. An ideal structure [15] of the unit cell is shown in Fig. 10a where displacement of AI ions is ignored and O ions are omitted. Figs. 10b and 10c show the [110] and [001] projections of the ideal structure, respectively. It is apparent from the comparison of Figs. 7 and 10b that the bright dots in the H R E M image correspond to columns of A1 vacancies. The distortion of the image around the dark- or brightline is quite small and limited to a narrow region as seen in Fig. 7. Furthermore, any shift of (112)
C. Kinoshita et al. / Ultramicroscopy 56 (1994) 2 1 6 - 2 2 4
223
[ ( ~ I- }- 1 ?'---:.~-"---' - .... ~13 i i
(e)
~I0]T
i(1T4)
~0=0.238 n m
A3 '
,~ i~i
Go
"-
!p
e
•
Q '
'
A'~"m~_o_[_~ ~Az
2
Fig. 10. Structure of perfect a-Al203. The arrangements of Al ions (closed cirles) and vacancies (open circles) are shown but the displacement of Al ions along c-axis is ignored_ O ions are omitted. (a) Three-dimensional view; (b) projection view along the [110] direction; and (c) Al layer of the (001) basal plane.
lattice f r i n g e s c a n n o t b e o b s e r v e d at t h e p l a n a r defect. H e n c e , it is u n l i k e l y t h a t t h e p l a n a r d e f e c t is a s t a c k i n g f a u l t c o n s i s t i n g o f excess or d e f i c i e n t
Ca) 13 Y Qtflev
(001) p l a n e s [16]. A n A l c o l u m n a l o n g t h e [001] d i r e c t i o n consists o f 2 / 3 A l ions a n d 1 / 3 A1 v a c a n c i e s as s h o w n in Fig. 10a . T h e A l c o l u m n s
c~ 13' y a
(b)
• oO • O o o O o p O o p 0 •
pOp
•
O o o O o oOo o 0 • pOp o O O o O o o O o e 0
•
pop
•
Ooo Oo • Oo • 0 o o O e • Oo gO • pOP go gOOD • O o o O o • Oo • O • • 0 • o O o o O o • O o o O • • 0 • o ~ C o
~ ~ ~ O o /
m
After rearrangement ~
o o 0 ~ • o • 0 o
~oO •
•
•
Cc)
OOO • • 0 O D D
O 0 • ~ O 0 •
•
•
p o e • • 0 t def ect
Original arrangement
Fig. 11_ Model of the planar defect in a-Al203. (a) The [110] projection; (o) columns consisting of AI ions alone, shaded circles indicate (o) columns consisting of Al vacancies alone, (~) columns of 50% Al ions and 50% Al vacancies. (b) The rearranged structure in the defect plane. (c) The original arrangement in the perfect plane; (o) AI ions, (O) Al vacancies.
224
C Kinoshita et aL / Ultramicroscopy 56 (1994) 216-224
are visible as b r i g h t dots in the [001] H R E M i m a g e o f Fig. 8. T h e p e r i o d i c m o d u l a t i o n in bright dots s e e m s to be d u e to t h e c h a n g e o f a t o m i c a r r a n g e m e n t s in an AI layer o f the b a s a l p l a n e . W e a s s u m e from t h e above discussion t h a t the p l a n a r d e f e c t is a result of r e a r r a n g e m e n t s of AI ions and A1 v a c a n c i e s in t h e (001) b a s a l p l a n e . A m o d e l of the r e a r r a n g e m e n t is shown in Fig. l l a . In the [110] p r o j e c t i o n of the p l a n a r defect, c o l u m n s of AI vacancies a l o n e a r e lost a n d t h o s e consisting of 50% of A1 ions a n d A1 v a c a n c i e s a r e p r e s e n t , w h e r e a s c o l u m n s of AI- vacancies ( o p e n circles) are a r r a n g e d with every o t h e r two c o l u m n s of AI ions (closed circles) in t h e p e r f e c t structure. This m e a n s that the ratio o f the n u m b e r of vacancies to t h e n u m b e r of AI ions in t h e d e f e c t p l a n e c h a n g e s f r o m 1 / 2 to 1 / 3 in a p e r f e c t plane. A n a t o m i c a r r a n g e m e n t in t h e d e f e c t p l a n e is shown in Fig. l l b along with the o n e in t h e p e r f e c t plane. A row of AI ions a l o n e a p p e a r s along t h e [110] d i r e c t i o n a n d t h e i d e n t i c a l d i r e c t i o n s with every o t h e r row which consists of 50% A1 ions a n d 50% AI vacancies. AI ions a n d A1 vacancies a r e a r r a n g e d so t h a t the s t r u c t u r e m a y have sixfold s y m m e t r y a n d a d o u b l e p e r i o d c o m p a r e d with t h e original structure. T h e m o d e l of t h e p l a n a r d e f e c t shown h e r e can qualitatively acc o u n t for the s t r e a k i n g and t h e extra spots in Fig. 9. It s h o u l d be p o i n t e d o u t t h a t t h e ratio of the n u m b e r of AI v a c a n c i e s to the a t o m i c sites in the d e f e c t AI p l a n e is e q u a l to o n e f o u r t h which c o r r e s p o n d s to t h e c o m p o s i t i o n of A 1 3 0 2, while the r a t i o in t h e p e r f e c t A1 p l a n e in a - A 1 2 0 3 is one third. T h e r e a r r a n g e d s t r u c t u r e shown in Fig. l l b is actually the s a m e as t h e o n e of [111] layers of 7 - A l 3 0 4 (spinel-type). It is n o t i c e a b l e t h a t Bursill a n d P e n g Ju Lin [17] i n t e r p r e t e d , from c o m p u t e r s i m u l a t i o n of [110] H R E M images, the d a r k - l i n e s as b e i n g d u e to facets t e r m i n a t e d with a m o n o l a y e r 7-A1304 structure. T h e s e results suggest t h e local r e d u c t i o n of A1203 on a m o n o layer scale. W e have, however, not d e t e c t e d any sign in H R E M i m a g e s a n d diffraction p a t t e r n s to i n d i c a t e t h e p r e s e n c e of 7-A1304 p h a s e or m e t a l lic A1 phase. O n t h e o t h e r hand, Bonevich a n d M a r k s [6] o b s e r v e d small crystallites of A1 at
surface of a - A l 2 0 3 i n s t e a d of t h e d a r k - l i n e s w h e n the s p e c i m e n was i r r a d i a t e d u n d e r u l t r a - h i g h vacuum. This result also i n d i c a t e s t h a t the d a m a g e p r o c e s s is very sensitive to t h e d e g r e e of v a c u u m a r o u n d the s p e c i m e n .
4. Conclusion N e w h i g h - r e s o l u t i o n a n d h i g h - v a c u u m transmission e l e c t r o n m i c r o s c o p e s r e v e a l r a d i a t i o n effects m o r e clearly a n d t h e y are g o o d facilities for e s t a b l i s h i n g t h e m e c h a n i s m s of effects, which a r e i n d i s p e n s a b l e to o b t a i n i n g q u a n t i t a t i v e s t r u c t u r a l and c h e m i c a l i n f o r m a t i o n of m a t e r i a l s .
References [1] C. Kinoshita, J. Electron Microsc_ 34 (1985) 299. [2] N. Itoh, Adv. Phys. 31 (1982) 491. [3] F.W. Clinard Jr. and L.W. Hobbs, Radiation effects in non-metals, in: Physics of Radiation Effects in Crystals, Eds. R.A. Johnson and A.N. Orlov (North-Holland, Amsterdam, 1986) p. 387. [4] M_ Kiritani and H. Takata, J. Nucl. Mater. 69/70 (1978) 277. [5] Y. Tomokiyo, T. Kuroiwa and C_ Kinoshita, Ultramicroscopy 39 (1991) 213. [6] J.E. Bonevich and L.D. Marks, Ultramicroscopy 35 (1991) 161. [7] E.P. Butler and J.F. Orchard, An experimental investigation of electron irradiation-induced disordering mechanisms, in: Phase Stability During Irradiation, Eds. J.R. Holland, L_K- Mansur and D.I. Potter (Met. Soc. AIME, New York, 1981) p. 315_ [8] T. Mukai, C. Kinoshita and S. Kitajima, Phil. Mag. A 47 (1983) 255. [9] K. Nakai and C. Kinoshita, J. Nucl. Mater. 169 (1989) 116. [10] J.W. Cahn. and J.E. Hilliard, J. Chem. Phys. 28 (1958) 2581. [11] J.W. Cahn, Acta Metall. 10 (1962) 179. [12] S. Horiuchi and Y. Matsui, J. Cryst. Soc. Jpn. 25 (1983) 3. [13] L.A. Bursill, Peng Ju Lin and D.J. Smith, Ultramicroscopy 23 (1987) 223. [14] P.S. Turner, T.J_ Bullough, R.W. Devenish, D.M. Maher and C.J. Humphreys, Phil. Mag. Lett. 61 (1990) 181. [15] M.L. Kronberg, Acta Metall. 5 (1957) 507. [16] A.Y. Stathopoulos and G.P. Pells, Phil. Mag. A 47 (1983) 381. [17] L.A. Bursill and Peng Ju Lin, Phil. Mag. A 60 (1989) 307.