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Some aspects on the structure of Guinier-Preston zones in AlCu alloys based on high resolution electron microscope observations
Scripta METALLURGICA
Vol. 22, pp. 947-951, 1988 Printed in the U.S.A.
Pergamon Press plc All rights reserved
VIEWPOINT SET No. 13
SOMEASPECTS ON THE STRUCTUREOF GUINIER-PRESTON ZONES IN AI-Cu ALLOYS BASED ON HIGH RESOLUTION ELECTRONMICROSCOPEOBSERVATIONS H. Yoshida Research Reactor Institute, Kyoto University, Kumatori-cho, Osaka 590-04, Japan
(Received March 17, 1988) (Revised April 27, 1988) Introduction Since the discovery of G.P. zones in Al-Cu alloys by Guinier (1) and Preston (2) in 1938, the structure of these zones has been studied by many investigators and i t has becomean interesting research subject in the f i e l d s of metallurgy and crystalography. Recent progress of experimental techniques and equipments which allow optical imaging with high resolution lead us to have more detailed information and to discuss the structure of G.P. zones on an atomic scale. High resolution electron microscopy, as one of the most powerful techniques, has already given atomic resolution images of crystals containing G.P. zones which include important information about the atomic arrangements around G.P. zones in Al-Cu alloys (3-8). In this paper the essence of our previous observations on an AI-3.9 wt.% Cu (1.74 at.% Cu) alloy using the weak-beamand the multi-beam techniques are reviewed, discussed and comparedwith the structure models. Structure of G.P.(1) Zones An example for atomic resolution of an electron microscopic image of G.P. zones is shown in Fig. 1 for the [001] oriented crystal agedfor 10.8 ks at 413 K followed by 2146.4 ks at room temperature. The multi-beam images were obtained at various defocus values when the optical axis was set on the center direction between the primary beam and the (200), (020) and (220) reflections. Under these optimum conditions clear images of l a t t i c e s are obtained which are distorted by G.P. zones (9). In Fig. 1, the bright dots have been interpreted as images of rows of Al atoms positioned in the [001] direction normal to the f o i l plane in a thin section of the crystal (4,5). The arrays of dotted images with stronger brightness indicated by A, C, D and E correspond to planes of Cu atoms of G.P. I zones lying on the (200) and (020) planes parallel to the incident electron beam. The brighter dotted-images surrounded by dark contrast on either side of the arrays were interpreted by contrast calculations based on the dynamic diffraction theory. The model consists of a monolayer of Cu atoms which induces displacements of the surrounding Al l a t t i c e atoms (5,9). I t is based on the same assumption as in a previous paper (10), namely that the Cu layer behaves as a dislocation loop with a low Burgers vector which is responsible for the distortion f i e l d in the Al l a t t i c e . Its magnitude is f i t t e d to match the experimental observations. For the contrast calculation by the multi-slice method, however, the atomic displacement at the model is assumed to be independent on the z direction. The model is good enough to explain the multi-beam images of G.P.(1) zones. From the appearanceof the l a t t i c e image i t can be concluded that there exists a farreaching displacement f i e l d in the v i c i n i t y of the Cu layer. Sometimesi t is influenced by strain f i e l d s of other zones. For example, the displacement f i e l d continues from zone A to zone E (8). Sometimes, an asymmetric displacement f i e l d was also observed as in zone D, while zone C shows a symmetric f i e l d (8). The l a t t e r seems to be the standard structure (11). At positions F and G the investigation of imageswith other defocus values led to the conclusion that these images consist of the overlap of two G.P.I zones which have formed near the top or bottom in the f o i l on different (200) planes which have a distance of two (200) layers. Such image overlapping of two G.P.(1) zones sometimes appears when the f o i l is r e l a t i v e l y thick. There is no reasonable image of a G.P.(I) zone showing multiple Cu layers (n-layers).
947
948
STRUCTURE OF G-P ZONES
Vol.
22, No.
7
FIG. 1. Atomic resolution electron microscope image of Al-3.97wt%Cu alloy containing G.P. zones. A, C, D and E indicate mono-layer G.P.(1) zones. B, M and N indicates doublelayer G . P . ( I I ) zones. F and G see text.
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Vol.
22, No.
7
S T R U C T U R E OF G-P ZONES
949
The model of a G.P.(1) zone (Fig. 2) used in this paper contains larger atomic displacements and a wider displacement f i e l d than the model suggested by Gerold (12}. The top and bottom numbers in the spheres represent the atomic displacement in the y and x directions, respectively, given in units of ao/lO0 where ao is the l a t t i c e parameter (10). This structure suggests that the effective atomic size of Cu atoms in the zone is s l i g h t l y smaller than the regular size of Cu, which might be caused by an electronic structure producing two-dimensional Cu-Cu bonds in the Al l a t t i c e . This may also be the reason for the formation of the monolayer structure of Cu atoms avoiding multiple layers and a three-dimensional arrangement. There seem to be no arguments for the existence of multi-layer zones of Cu atoms after low temperature aging. As the model in Fig. 2 shows there exists a l a t t i c e compression in the y direction in the Al layer adjacent to the end of the Cu layer. In these areas there may also be the sites for vacancies which enhance the diffusion of Cu atoms from the matrix to the border of the Cu layer (7). When after reversion for 60 s at 448 K the AI-3.g?wt%Cu alloy was aged for 5184 ks at room temperature small mono-layerG.P.(I) zones were formed with a high density, while after flash-annealing for 60 s at 403 K an aging for 8900 ks at room temperature leads to a low density of G.P.(1) zones with a large average diameter (13.5 nm). This size is much bigger than that formed during the usual aging at temperatures between 383 and 413 K (11). All these facts suggest the monolayer of Cu atoms as the standard structure for G.P.(1) zones rather than a multiple layer structure. For higher aging temperaturesthere may exist p o s s i b i l i t i e s for forming multi-layer structures. Structure of G.P.(II) Zones In electron microscopic observations using the weak-beam and the multi-beam methods, G.P.(1) and ( I I ) zones have been commonly observed to coexist in the early stages of aging in Al-Cu alloys (3,4). For example, G.P.(II) zones are also seen in Fig. 1. There are the images of double-layers of Cu atoms as indicated by M and N. The two Cu layers are separated by two and three Al layers for the image M and N, respectively. The image N shows the familar atomic arrangement with the standard structure of G.P.(II) zone (10,11) which consists of double-layers of Cu atoms separated by three (020) planes. The image M shows a deviation from this standard structure, where the atomic arrangement corresponds more to a layer of the intermediate 8' phase having the thickness of half a l a t t i c e constant in the c direction. The complicated image B in Fig. 1 can be roughly interpreted as a double-layer structure of G.P.(II) zone type with additional shear displacements as i t is found by contrast calculations (5,9). Both models of the G.P.(II) zone with or without a shear displacementwere constructed in such a way that two layers of Cu atoms were separated by three Al layers giving similar atomic displacements of the surrounding Al l a t t i c e as in the model of the G.P.(1) zone (9). When the size of G.P. zones was measured from weak-beammicrographs, the size distributions of G.P.(1) and ( I I ) zones could be deduced as shown in Figs. 3 (a) and (b) for samples aged for 149.4 ks at 408 K and for 864 ks at 413 K, respectively (11). The shaded areas correspond to G.P.(II) zones, and the others are G.P.(1) zones (index 1). The double-layer (dotted areas, index 2) and multi-layer G.P.(II) zones (indices 3 to 10) are larger in the average size than the G.P.(1) zones, but the size distributions overlap each other. In the growth process of the zones the extra Cu atoms supplied near the edge of the f i r s t layer may form a nuclei there for the second layer of Cu atoms. From Fig. 3 (a) i t is suggested that the probability to nucleate the second layer increases when the f i r s t layer becomes larger than a certain c r i t i c a l size, but this size is not unique because i t depends on several factors as vacancy concentration, temperature, strain f i e l d , etc. Fig. 3 (b) shows an example of the size distribution for the multilayer G.P.(II) zones after prolonged aging for 864 ks at 413 K. Since a size distribution can be measured for each Cu layer (11) the size distribution of the central layer is plotted on the figure. Up to ten layers could be observed on the micrographs. The result indicates that a l l these multi-layers of G.P. zones coexist with each other and that their size distributions overlap showing almost a normal distribution function. This type of size distribution is commonlyobserved for each Cu layer even when two structures coexist in the early stages of aging. I f the double-layer structure separated by three Al layers is defined as the G.P.(II) zone, we can recognize somedifferent features in the structure for the multi-layers, i . e . , the 8" phase (7,11). However, there is no reason to distinguish a G.P.(II) zone as a d i s t i n c t stage from that of the B" phase in the aging process. Also, there exists no other reason to d i s t i n guish the stage of the G.P.(II) from that of the G.P.(I) zone. From the observations i t is
950
STRUCTURE
OF G-P ZONES
Vol.
22, No.
7
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FIG. 3. Distributions in size and structure for G.P. zones in AI-3.9wt%Cu alloy aged for 149.4 ks at 408 K (a) and for 864 ks at 413 K (b). N and n are total and fractional numbers of G.P. zones, respectively, d and dm correspond to average diameters of whole of concluded that the s t r u c t u r a l changes from mono-layer G.P.(I) zones to double-layer G . P . ( I I ) zones and then to m u l t i l a y e r structures (9") is the ideal standard sequence f o r the p r e c i p i t a t i o n process (7,11). However, the images observed occasionally suggest the existence of deviated structures, although these structures always are not complete (11). I f , f o r example, the double-layer s t r u c ture separated only by two AI layers is stacked to form a m u l t i - l a y e r s t r u c t u r e , the r e s u l t i n g p e r i o d i c i t y produces the same ordered structure as that of the 8' phase and t h i s f a c t may suggest a p o s s i b i l i t y to form t h i s phase in the growth process. Structural Modifications Under I r r a d i a t i o n Conditions Structural modifications occurring during aging under radiation conditions are much larger compared to those observed during normal aging. An example is given in Fig. 4, where the s t r u c t u r a l changes during natural aging under 1.25 MeV electron i r r a d i a t i o n are p l o t t e d as function of the aging ( i . e . , i r r a d i a t i o n ) time. The sample was pre-aged f o r 360 ks at 413 K where G.P.(I) zones had already formed. The f o l l o w i n g d i f f u s i o n - c o n t r o l l e d k i n e t i c s are much f a s t e r due to radiation-induced vacancies. Besides the average zone diameter D the Cu content CCu (at.% Cu) an~ the ~ensity p of the zones are plotted in the f i g u r e . During the i n i t i a l low dose rate of 101° e/cm~s a s l i g h t increase of D and a larger increase of p and CCu is observed followed by a decrease of the l a t t e r two parameters. This indicates a beginning d i s s o l u t i o n of G.P. zones. A f t e r the i r r a d i a t i o n rate of the electron beam has been increased to 1019 e/cm2s, the size D of the G.P.(I) zones decreases r a p i d l y i n d i c a t i n g a f u r t h e r d i s s o l u t i o n of these zones. At the same time G . P . ( I I ) zones are developed which grow r a p i d l y to a maximum size followed by a decrease as seen in Fig. 4. Due to the enhanced d i f f u s i o n s t r u c t u r a l changes toward the more complex structures occur at a much higher rate. Probably, these structures may contain more defects compared to those occurring during normal aging. The Cu concentration CCu ( i . e . , the t o t a l number of Cu atoms present in a l l G.P. zones) is always decreasing showing the r a d i a t i o n induced d i s s o l u t i o n which occurs simultaneously during the aging process.
Vol.
22, No.
7
STRUCTURE
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FIG. 4. Changes of average diameter ~ and density p of G.P. zones in an Al-3.97wt%Cu alloy during an in-situ HVEMexperiment. CCu (at.Cu) indicates the amount of Cu atoms precipitated to the zones. Cv is the ratio of numbers of vacancies contained in clusters to those produced in Frenkel pairs. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
A. Guinier, Nature, 142, 569 (1938); Comp. Ren., 206, 1641, 1972 (1938). G.D. Preston, Nature, 142, 570 (1938); Proc. Roy. Soc., A167, 526 (1938). H. Yoshida, H. Hashimoto and Y. Yokota, Electron Microscopy 1978, Vol. 1, p. 306, (1978); Ann. Repts Res. Reactor Inst. Kyoto Univ., 13, 208 (1980). H. Yoshida, H. Hashimoto and Y. Yokota, Electron Microscopy 1980, Vol. 1, p. 268, Europ. Cong. Elect. Micros. Fund., The Hague, (1980). H. Yoshida, H. Hashimoto, Y. Yokota and N. Ajika, Trans. Jap. Inst. Metals, 24, 378 (1983). H. Yoshida, H. Hashimoto, Y. Yokota and M. Takeda, Symp. Proc. Vol. 21, p. 131, MRS, (1984). H. Yoshida, Decomposition of Alloys: the early stages, p. 191, Pergamon Press, (1984). T. Sago and T. Takahasi, Trans. Japan Inst. Metals, 24, 386 (1983). N. Ajika, H. Endoh, H. Hashimoto, M. Tomita and H. Yoshida, Phil. Mag. A, 51, 729 (1985). H. Yoshida, D.J.H. Cockayne and M.J. Whelan, Phil. Mag., 34, 89 (1976). H. Yoshida, Phase Transformation, p. 363, (1982). V. Gerold, Z. Metallkde., 45, 593, 599 (1954); Acta Cryst. 11, 230 (1958).
Vol. 22, pp. 947-951, 1988 Printed in the U.S.A.
Pergamon Press plc All rights reserved
VIEWPOINT SET No. 13
SOMEASPECTS ON THE STRUCTUREOF GUINIER-PRESTON ZONES IN AI-Cu ALLOYS BASED ON HIGH RESOLUTION ELECTRONMICROSCOPEOBSERVATIONS H. Yoshida Research Reactor Institute, Kyoto University, Kumatori-cho, Osaka 590-04, Japan
(Received March 17, 1988) (Revised April 27, 1988) Introduction Since the discovery of G.P. zones in Al-Cu alloys by Guinier (1) and Preston (2) in 1938, the structure of these zones has been studied by many investigators and i t has becomean interesting research subject in the f i e l d s of metallurgy and crystalography. Recent progress of experimental techniques and equipments which allow optical imaging with high resolution lead us to have more detailed information and to discuss the structure of G.P. zones on an atomic scale. High resolution electron microscopy, as one of the most powerful techniques, has already given atomic resolution images of crystals containing G.P. zones which include important information about the atomic arrangements around G.P. zones in Al-Cu alloys (3-8). In this paper the essence of our previous observations on an AI-3.9 wt.% Cu (1.74 at.% Cu) alloy using the weak-beamand the multi-beam techniques are reviewed, discussed and comparedwith the structure models. Structure of G.P.(1) Zones An example for atomic resolution of an electron microscopic image of G.P. zones is shown in Fig. 1 for the [001] oriented crystal agedfor 10.8 ks at 413 K followed by 2146.4 ks at room temperature. The multi-beam images were obtained at various defocus values when the optical axis was set on the center direction between the primary beam and the (200), (020) and (220) reflections. Under these optimum conditions clear images of l a t t i c e s are obtained which are distorted by G.P. zones (9). In Fig. 1, the bright dots have been interpreted as images of rows of Al atoms positioned in the [001] direction normal to the f o i l plane in a thin section of the crystal (4,5). The arrays of dotted images with stronger brightness indicated by A, C, D and E correspond to planes of Cu atoms of G.P. I zones lying on the (200) and (020) planes parallel to the incident electron beam. The brighter dotted-images surrounded by dark contrast on either side of the arrays were interpreted by contrast calculations based on the dynamic diffraction theory. The model consists of a monolayer of Cu atoms which induces displacements of the surrounding Al l a t t i c e atoms (5,9). I t is based on the same assumption as in a previous paper (10), namely that the Cu layer behaves as a dislocation loop with a low Burgers vector which is responsible for the distortion f i e l d in the Al l a t t i c e . Its magnitude is f i t t e d to match the experimental observations. For the contrast calculation by the multi-slice method, however, the atomic displacement at the model is assumed to be independent on the z direction. The model is good enough to explain the multi-beam images of G.P.(1) zones. From the appearanceof the l a t t i c e image i t can be concluded that there exists a farreaching displacement f i e l d in the v i c i n i t y of the Cu layer. Sometimesi t is influenced by strain f i e l d s of other zones. For example, the displacement f i e l d continues from zone A to zone E (8). Sometimes, an asymmetric displacement f i e l d was also observed as in zone D, while zone C shows a symmetric f i e l d (8). The l a t t e r seems to be the standard structure (11). At positions F and G the investigation of imageswith other defocus values led to the conclusion that these images consist of the overlap of two G.P.I zones which have formed near the top or bottom in the f o i l on different (200) planes which have a distance of two (200) layers. Such image overlapping of two G.P.(1) zones sometimes appears when the f o i l is r e l a t i v e l y thick. There is no reasonable image of a G.P.(I) zone showing multiple Cu layers (n-layers).
947
948
STRUCTURE OF G-P ZONES
Vol.
22, No.
7
FIG. 1. Atomic resolution electron microscope image of Al-3.97wt%Cu alloy containing G.P. zones. A, C, D and E indicate mono-layer G.P.(1) zones. B, M and N indicates doublelayer G . P . ( I I ) zones. F and G see text.
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Vol.
22, No.
7
S T R U C T U R E OF G-P ZONES
949
The model of a G.P.(1) zone (Fig. 2) used in this paper contains larger atomic displacements and a wider displacement f i e l d than the model suggested by Gerold (12}. The top and bottom numbers in the spheres represent the atomic displacement in the y and x directions, respectively, given in units of ao/lO0 where ao is the l a t t i c e parameter (10). This structure suggests that the effective atomic size of Cu atoms in the zone is s l i g h t l y smaller than the regular size of Cu, which might be caused by an electronic structure producing two-dimensional Cu-Cu bonds in the Al l a t t i c e . This may also be the reason for the formation of the monolayer structure of Cu atoms avoiding multiple layers and a three-dimensional arrangement. There seem to be no arguments for the existence of multi-layer zones of Cu atoms after low temperature aging. As the model in Fig. 2 shows there exists a l a t t i c e compression in the y direction in the Al layer adjacent to the end of the Cu layer. In these areas there may also be the sites for vacancies which enhance the diffusion of Cu atoms from the matrix to the border of the Cu layer (7). When after reversion for 60 s at 448 K the AI-3.g?wt%Cu alloy was aged for 5184 ks at room temperature small mono-layerG.P.(I) zones were formed with a high density, while after flash-annealing for 60 s at 403 K an aging for 8900 ks at room temperature leads to a low density of G.P.(1) zones with a large average diameter (13.5 nm). This size is much bigger than that formed during the usual aging at temperatures between 383 and 413 K (11). All these facts suggest the monolayer of Cu atoms as the standard structure for G.P.(1) zones rather than a multiple layer structure. For higher aging temperaturesthere may exist p o s s i b i l i t i e s for forming multi-layer structures. Structure of G.P.(II) Zones In electron microscopic observations using the weak-beam and the multi-beam methods, G.P.(1) and ( I I ) zones have been commonly observed to coexist in the early stages of aging in Al-Cu alloys (3,4). For example, G.P.(II) zones are also seen in Fig. 1. There are the images of double-layers of Cu atoms as indicated by M and N. The two Cu layers are separated by two and three Al layers for the image M and N, respectively. The image N shows the familar atomic arrangement with the standard structure of G.P.(II) zone (10,11) which consists of double-layers of Cu atoms separated by three (020) planes. The image M shows a deviation from this standard structure, where the atomic arrangement corresponds more to a layer of the intermediate 8' phase having the thickness of half a l a t t i c e constant in the c direction. The complicated image B in Fig. 1 can be roughly interpreted as a double-layer structure of G.P.(II) zone type with additional shear displacements as i t is found by contrast calculations (5,9). Both models of the G.P.(II) zone with or without a shear displacementwere constructed in such a way that two layers of Cu atoms were separated by three Al layers giving similar atomic displacements of the surrounding Al l a t t i c e as in the model of the G.P.(1) zone (9). When the size of G.P. zones was measured from weak-beammicrographs, the size distributions of G.P.(1) and ( I I ) zones could be deduced as shown in Figs. 3 (a) and (b) for samples aged for 149.4 ks at 408 K and for 864 ks at 413 K, respectively (11). The shaded areas correspond to G.P.(II) zones, and the others are G.P.(1) zones (index 1). The double-layer (dotted areas, index 2) and multi-layer G.P.(II) zones (indices 3 to 10) are larger in the average size than the G.P.(1) zones, but the size distributions overlap each other. In the growth process of the zones the extra Cu atoms supplied near the edge of the f i r s t layer may form a nuclei there for the second layer of Cu atoms. From Fig. 3 (a) i t is suggested that the probability to nucleate the second layer increases when the f i r s t layer becomes larger than a certain c r i t i c a l size, but this size is not unique because i t depends on several factors as vacancy concentration, temperature, strain f i e l d , etc. Fig. 3 (b) shows an example of the size distribution for the multilayer G.P.(II) zones after prolonged aging for 864 ks at 413 K. Since a size distribution can be measured for each Cu layer (11) the size distribution of the central layer is plotted on the figure. Up to ten layers could be observed on the micrographs. The result indicates that a l l these multi-layers of G.P. zones coexist with each other and that their size distributions overlap showing almost a normal distribution function. This type of size distribution is commonlyobserved for each Cu layer even when two structures coexist in the early stages of aging. I f the double-layer structure separated by three Al layers is defined as the G.P.(II) zone, we can recognize somedifferent features in the structure for the multi-layers, i . e . , the 8" phase (7,11). However, there is no reason to distinguish a G.P.(II) zone as a d i s t i n c t stage from that of the B" phase in the aging process. Also, there exists no other reason to d i s t i n guish the stage of the G.P.(II) from that of the G.P.(I) zone. From the observations i t is
950
STRUCTURE
OF G-P ZONES
Vol.
22, No.
7
(b)
(a)
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FIG. 3. Distributions in size and structure for G.P. zones in AI-3.9wt%Cu alloy aged for 149.4 ks at 408 K (a) and for 864 ks at 413 K (b). N and n are total and fractional numbers of G.P. zones, respectively, d and dm correspond to average diameters of whole of concluded that the s t r u c t u r a l changes from mono-layer G.P.(I) zones to double-layer G . P . ( I I ) zones and then to m u l t i l a y e r structures (9") is the ideal standard sequence f o r the p r e c i p i t a t i o n process (7,11). However, the images observed occasionally suggest the existence of deviated structures, although these structures always are not complete (11). I f , f o r example, the double-layer s t r u c ture separated only by two AI layers is stacked to form a m u l t i - l a y e r s t r u c t u r e , the r e s u l t i n g p e r i o d i c i t y produces the same ordered structure as that of the 8' phase and t h i s f a c t may suggest a p o s s i b i l i t y to form t h i s phase in the growth process. Structural Modifications Under I r r a d i a t i o n Conditions Structural modifications occurring during aging under radiation conditions are much larger compared to those observed during normal aging. An example is given in Fig. 4, where the s t r u c t u r a l changes during natural aging under 1.25 MeV electron i r r a d i a t i o n are p l o t t e d as function of the aging ( i . e . , i r r a d i a t i o n ) time. The sample was pre-aged f o r 360 ks at 413 K where G.P.(I) zones had already formed. The f o l l o w i n g d i f f u s i o n - c o n t r o l l e d k i n e t i c s are much f a s t e r due to radiation-induced vacancies. Besides the average zone diameter D the Cu content CCu (at.% Cu) an~ the ~ensity p of the zones are plotted in the f i g u r e . During the i n i t i a l low dose rate of 101° e/cm~s a s l i g h t increase of D and a larger increase of p and CCu is observed followed by a decrease of the l a t t e r two parameters. This indicates a beginning d i s s o l u t i o n of G.P. zones. A f t e r the i r r a d i a t i o n rate of the electron beam has been increased to 1019 e/cm2s, the size D of the G.P.(I) zones decreases r a p i d l y i n d i c a t i n g a f u r t h e r d i s s o l u t i o n of these zones. At the same time G . P . ( I I ) zones are developed which grow r a p i d l y to a maximum size followed by a decrease as seen in Fig. 4. Due to the enhanced d i f f u s i o n s t r u c t u r a l changes toward the more complex structures occur at a much higher rate. Probably, these structures may contain more defects compared to those occurring during normal aging. The Cu concentration CCu ( i . e . , the t o t a l number of Cu atoms present in a l l G.P. zones) is always decreasing showing the r a d i a t i o n induced d i s s o l u t i o n which occurs simultaneously during the aging process.
Vol.
22, No.
7
STRUCTURE
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FIG. 4. Changes of average diameter ~ and density p of G.P. zones in an Al-3.97wt%Cu alloy during an in-situ HVEMexperiment. CCu (at.Cu) indicates the amount of Cu atoms precipitated to the zones. Cv is the ratio of numbers of vacancies contained in clusters to those produced in Frenkel pairs. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
A. Guinier, Nature, 142, 569 (1938); Comp. Ren., 206, 1641, 1972 (1938). G.D. Preston, Nature, 142, 570 (1938); Proc. Roy. Soc., A167, 526 (1938). H. Yoshida, H. Hashimoto and Y. Yokota, Electron Microscopy 1978, Vol. 1, p. 306, (1978); Ann. Repts Res. Reactor Inst. Kyoto Univ., 13, 208 (1980). H. Yoshida, H. Hashimoto and Y. Yokota, Electron Microscopy 1980, Vol. 1, p. 268, Europ. Cong. Elect. Micros. Fund., The Hague, (1980). H. Yoshida, H. Hashimoto, Y. Yokota and N. Ajika, Trans. Jap. Inst. Metals, 24, 378 (1983). H. Yoshida, H. Hashimoto, Y. Yokota and M. Takeda, Symp. Proc. Vol. 21, p. 131, MRS, (1984). H. Yoshida, Decomposition of Alloys: the early stages, p. 191, Pergamon Press, (1984). T. Sago and T. Takahasi, Trans. Japan Inst. Metals, 24, 386 (1983). N. Ajika, H. Endoh, H. Hashimoto, M. Tomita and H. Yoshida, Phil. Mag. A, 51, 729 (1985). H. Yoshida, D.J.H. Cockayne and M.J. Whelan, Phil. Mag., 34, 89 (1976). H. Yoshida, Phase Transformation, p. 363, (1982). V. Gerold, Z. Metallkde., 45, 593, 599 (1954); Acta Cryst. 11, 230 (1958).