- Email: [email protected]
A method for simulating electron microscope dislocation images
Materials Science and Engineering, A164 (1993) 373-378
373
A method for simulating electron microscope dislocation images R. Sch~iublin lnstitut de GOnie Atomique, Ecole Polytechnique F~d&ale de Lausanne, CH-IOI5 Lausanne (Switzerland)
P. Stadelmann Institut Interd~partemental de Microscopic Electronique, Ecole Polytechnique Fdd&ale de Lausanne, CH- I OI 5 Lausanne (Switzerland)
Abstract An image simulation program has been developed to quantitatively evaluate transmission electron microscopy (TEM) images of dislocation configurations. It takes into account many beams, includes linear anisotropic elasticity and uses the column approximation. The program simulates the image of 1 to 4 parallel dislocations and their associated planar faults under bright field, dark field and various weak-beam TEM conditions. Simulations of a variety of dislocations are presented here and it is shown that the use of a many-beam calculation is essential in the case of weak-beam identification of weakly dissociated dislocations.
1. Introduction
The resolution offered by weak-beam T E M allows for the direct observation of small lattice defects such as dissociated dislocations [1, 2]. This may yield a better understanding of the mechanical properties, especially of certain intermetallic compounds such as Ni3A1, which exhibits peculiar mechanical properties [3-5] related to dissociated dislocations [6]. Unfortunately results from the weak-beam method cannot be directly related to dislocation separations. The position of the peak does not correspond exactly to the actual dislocation position, and furthermore, there can be additional peaks under certain T E M conditions. Cockayne et al. proposed corrections on peak positions assuming linear isotropic elasticity [7]. This paper describes a new program of image simulation based on the work of Head et al. [8]. It is used to define reliable simulation conditions for two-beam and weak-beam observations. It is also shown that accurate dislocation distances can be measured under weakbeam conditions, by using this program to calculate dislocation image shifts. Head et al. calculated the images of dislocations using the Howie-Whelan equations, that take into account the transmitted beam and one diffracted beam only. The so-called generalized cross-section method was applied to reduce the number of integrations. This concept, based on the fact that the deformation field around a straight dislocation has a cylindrical symmetry, restricts the program to the calculation of straight, parallel and infinite dislocations. 0921-5093/93/$6.00
The geometry of the generalized cross-section defines a configuration where the dislocations intersect the sample surfaces and cannot be parallel to them. Nevertheless, it has been shown [8] that the displacement field of an infinite crystal can be successfully used to describe a dislocation in a thin foil when surface relaxations are ignored. The program of Head et al. has been extended to calculate images of 1 to 4 dislocations and their associated planar faults. The development of a linear differential equation system describing the propagation of the electrons in the sample was made. It allows simulation of images using more than two beams. It has been estimated [8] that about 90% of the total intensity was contained in the first two beams and it was deduced that the two-beam simulation was suitable for many situations. However, when using kinematic conditions, this may not be true any more and the need of a manybeam calculation may arise, especially in weak-beam conditions when the Ewald sphere intersects diffracted spots that are far from the transmitted spot. The intersected spot becomes excited and may contribute more strongly compared with diffracted spots which are closer to the transmitted spot. The number and type of beams that have to be taken into account in the calculation is therefore not obvious. They will be discussed in order to find the optimal simulation conditions, i.e. the simulation conditions where the number of beams is minimal to reduce the computing time together with a calculated image as alike as possible to real observation. © 1993 - Elsevier Sequoia. All fights reserved
374
R. Schiiublin, P. Stadelmann
/ Simulating electron microscope dislocation images
2. Theory To calculate the propagation of electrons in a faulted crystal, the dynamic theory of contrast (see Hirsch et al. [9]) is used. The crystal at a point r is described by a faulted potential V(r) that can be written as a Fourier series: V(r)=
[Ug exp(- 2:tigR(r))] exp(2~rigr)
beam
direction:/Z /z+g+Sg'~ 4g tO0/ ¢p,/
(1) Ewald sphere
-
The summation is done over all reciprocal lattice vectors g, with m e the electron mass and h the Planck constant. R(r) describes the displacement field around the lattice defect. The SchrSdinger's wave equation is: ~72tI/(r)+ ( ~ 2 me) [E + V(r)]W(I)=0
(2)
W(r) is the function associated with the electron wave that moves through the faulted crystal. The proposed solution to the SchrSdinger equation is: W ( r ) = 7 q~ exp(2xi(z +g+Sg)r)
The matrix M is symmetrical: A
A I
AI
BI
• ..
A i
0q~(r) = Z zri 0~/~ - h ~g-h qgh(r)exp[2zti(h- g)R(r) + 2:ri(Sh -- Sg)r]
(4)
The equations are to be integrated along the proper beam direction r/g, which is difficult to solve analytically. To simplify, we use the column approximation that allows integration along the same direction for all beams (6/6r/=d/dz). We can then write the equation system (4) in matrix form:
[Co(r)1 dq~(r)= Mq~(r)
where , ( r ) = [ qg'(r)[:
L q~£(r)j
(5)
...
A~"
C. Ai
Bi
Cii
".°
B.
An
The q0g(r)function is associated with the beams that come out of the sample, including the transmitted beam (beam 0) and the diffracted beams (beam 1 to n). The contrast intensity I recorded on the micrographs for a g image, is simply I = q0g(r).tpg*(r). Figure 1 shows the crystal and the beams in the reciprocal space. Z is the direction of the transmitted beam; g is the reciprocal lattice vector and sg the deviation parameter. (Z + g + Sg) is the direction of the beam with index g and we will take a co-ordinate r/g along this direction. When V(r) and W(r) are substituted in the Schr6dinger equation, a system of n differential equations of the first order with n unknowns qgg(r)is obtained:
dz
Fig. 1. Representation of a g(4g) weak beam.
(3)
g
;oYS2 ematic
~
1+ i/
A= Bi
¢0
(6)
f =L-
Cij = i~1
+'77-~i j
~,. defines the extinction distance which becomes large if beam i becomes far from the transmitted beam; ~'i is related to ~i through a material-dependent absorption constant. Equation (5) shows that the derivative of each beam is a linear combination of all the beams. The contribution of each beam in the linear combination is weighted by the matrix coefficients (6). The physical meaning of relation (5) is that the contribution of a beam i depends on the associated deviation parameter si (diffraction condition), the associated extinction distance ~i (material characteristics) and a cross-term ~i_j that has the expression of an extinction distance. High Igi] values correspond to large ~i and therefore beam gi has a weak contribution. Conversely, beam gi has a strong contribution when the deviation parameter sz is small. The importance of the contribution is a complex balance between the observation conditions s~ and the material characteristics (~). This has to be taken into account in the choice of the beams included in the
R. Schiiublin, P. Stadelmann
/
375
Simulating electron microscope dislocation images
calculation: they have to be close to the transmitted beam (small ~i) and with g~ situated inside and close to the Ewald sphere (smaller sg). The dependence on ~i / indicates that if we take for instance beam gg, then beams g~_~ and g~+~ will contribute strongly to beam gg as ~ i is small.
"cross-section"
thin foil with 1dislocation b=a[1, 1, 1] thickness : 155.52 nm
3. Simulations
In order to illustrate the capabilities of the program, different simulations were performed on dislocations in r-brass. This CuZn alloy exhibits strong anisotropy (A-- 8.5) and has a b.c.c, structure. The same material has been used by Head et al. [8] for various image simulations. We have compared our calculations with one of their examples by using the same conditions ( V ~ = 100 kV, C~ = 129, C12=110, C44=82 GPa). We then extended the calculations by including additional beams to the initial two beams. We consider successively the case of one dislocation (Fig. 2(a)) with g = [020] and two parallel dislocations (Fig. 2(b)) with g=[1, - 1 , 0]. In order to compare the contributions of the various beams, images of one dislocation have been calculated using an increasing number of beams. Figure 3 gives a series of simulated bright field images and two series of calculated weak beam images. Two different weakbeam conditions, g(3g) and g(4g), have been investigated. The image shift associated with neighboring dislocations can be corrected with comparisons to simulated images. Figure 4 presents calculated images of two dislocations which correspond to Fig. 2(b), with different distances of separation, and under dark-field and weak-beam conditions. The safest simulation conditions were chosen according to the results of Fig. 3. From the simulations it is obvious that the width of the image peaks is decreased by using weak-beam conditions and that the weak-beam condition is the most appropriate for observing dissociation distances.
4. Results and discussion
Figure 3(a) shows that in a bright field the image calculated with (0, g) does not change significantly through the addition of other beams, and indicates that the two-beam simulation conditions are suitable for bright field images. This simulated image is in satisfactory agreement with the one published by Head et al. [8]. These two points emphasize the reliability of the present program. As far as weak-beam images simulation are concerned, the first image of Fig. 3(b) and (c) has been calculated with five successive beams. We found that
b) foil ,norma
i i
i i
,
,.7;~.q'7.H:!..%~;.:i.;#~'s.~?~-~.',:ci.~:7.~7:~(~!+!cT,'~:'~.'~'U%'.'+~c."
• ÷:..: .....~..
~ .,:.,:e..~.
~
~ .-.:.:.~...:
......
:.~....:.~ .........
image
[1, 1, 3] thin foil with 2 dislocations in
(~,o, 1) bT,z=a[1, O,
1]
thickness : 55.3 nm image
Fig. 2. Simulation geometry: configurationwith (a) one dislocation, (b) two dislocations.
additional beams do not change the image significantly and are therefore considered as the most reliable ones. Simulations with less than five beams are presented in Fig. 3(b) and (c). For both weak-beam conditions, the image is not satisfactory when only the transmitted beam and the g beam are considered. For the g(3g) conditions, the image which corresponds to a four-beam calculation (0 to 3g) is identical to the reliable one. The image corresponding to three beams (0 to 2g) shows some similarities with the reliable one but with three other beams (0, g, 3g), it looks completely different. Therefore in this case, a rather reliable simulation requires at least three beams (0, g, 2g). For the g(4g) conditions, Fig. 3(c), the image which corresponds to a four-beam calculation (0-3g) is very reliable, while the others are not. The general rule that can be derived from these observations and, accordingly, to the above considerations on relation (5), is that the beams that have to be taken into account for the calculation are the following: the transmitted beam, the diffracted beam chosen to form the image and diffracted ones that are close to the Ewald sphere. It can be observed that the excited beam is not essential for the calculation. In what concerns dislocation separation measurements, Fig. 4(a) shows the dark field images are not appropriate for observing dislocation separation below 10 nm. Figure 4(b) suggests that the weak beam g(4g) is the most suitable condition for observing dislocations in fl-brass that are separated by distances close to
376
a) BF O(g)
R. Schiiublin, P. Stadelmann
/
Simulating electron microscope dislocation images
b) WB g(3g)
0 g
c) WB g(4g)
3g
0 to 4g
0 g 2g
0 g
0 g
0 g 2g 3g
0 g 2g
0 g 2g
-g to 3g
0 to 3g
-2g to 3g
0 g 3g
m m
m
0 to 3g
0 4g
Og 3g 4g
m
Fig. 3. Examples of simulated images of one dislocation. The beams used in the simulation are indicated on the left. (a) Bright field (two-beam case), (b) weak beam g(3g), (c) weak beam g(4g).
R. Schiiublin, P. Stadelmann
a) DF g(g)
/
Simulating electron microscope dislocation images
b) W B
377
c) WB g(4g)
d=5nm
2 nm
~g)
2 nm
5
g
5
nm
(2g)
nm
10
g
nm
(3g)
0 ~m
20 nm
74g )
40
40
nm
nm
1 I
Fig, 4. Examples of simulated images of two parallel dislocations. (a) Dark field (two-beam case): two beams (0, g). The separation distance is indicated on the left. (b) Weak beam: separation of 5 nm, six beams ( - g to 4g). The observation condition is indicated on the left. (c) g(4g) weak beam: six beams ( - g to 4g). The separation distance is indicated on the left.
5 nm, and Fig. 4(c) gives the calculated images for different dislocation distances. Figure 5 shows more quantitatively how the observed distance depends on the actual distance for a configuration close to Fig. 2(b) with b l = a [ 1 0 1 ] , b2=a[010], e - = [ - l 1 3 ] , g = [ l l 0 ] . For separations smaller than 10 TUn the correction is found to be important. It indicates that the image shift due to the displacement field overlap has to be taken into account in this case. The error introduced by the column approximation will only become important at
small distances [10]. Cockayne indicates a distance limit of about 0.5 nm in the g(4g) case [11].
5. Conclusion The development of a many-beam program to simulate T E M images of straight parallel dislocations in weak-beam conditions has been achieved.
378
R. Schiiublin, P. Stadelmann
/
Simulating electron microscope dislocation images
15
interest in this work. We also gratefully acknowledge the help of A. VtJser in calculating all the images appearing in this paper.
lO
References 5
o
5
lO
15
Fig. 5. Observed dislocations distance (nm) (simulation) as a function of the actual distance (nm). T h e use of many-beam calculations, as c o m p a r e d to two-beam calculations, appears to be necessary when imaging in weak-beam conditions, i.e. far from the Bragg condition. T h e program allows for the correction of the image shift through the comparison of T E M observed dislocation images and simulated images. Examples of this type of quantitative analysis are given in refs. 12 - 14.
Acknowledgments Support for this project was provided by Fonds National Suisse de la Recherche Scientifique. T h e authors would like to thank B. Viguier, Dr. N. Baluc, Dr. K. H e m k e r and Professor G. Vanderschaeve for fruitful discussions and Professor J. L. Martin for his
1 D. J. H. Cockayne, M. L. Jenkins and I. L. E Ray, Philos. Mag., 24 (1971) 1383. 2 M. L. Jenkins, Philos. Mag., 26 (1972) 747. 3 V. Paidar, D. P. Pope and V. Vitek, Acta Metall., 32 (1984) 435. 4 M. J. Mills, N. Baluc and H. P. Kamthaler, in C. T. Liu, A. I. Taub, N. S. Stoloff and C. C. Koch (eds.), High-Temperature Ordered Intermetallic Alloys III, MRS Pittsburgh, Pennsylvania, 1989, p. 203. 5 P.B. Hirsch, Philos. Mag. A, 65 (1992) 569. 6 V. Vitek, Dislocations and Properties of Real Materials, The Institute of Metals, London, 1985, p. 30. 7 D. J. H. Cockayne, I. L. F. Ray and M. J. Whelan, Philos. Mag., 20 (1969) 1265. 8 A.K. Head, P. Humble, L. M. Clarebrough, A. J. Morton and C. T. Forwood, Computed Electron Micrographs and Defect Identification, North-Holland Publishing Company, Amsterdam, 1973. 9 P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley and M. J. Whelan, Electron Microscopy of Thin Crystals, Butterworths, London, 1969. 10 A. Howie and C. H. Sworn, Philos. Mag., 22 (1970)861. 11 D.J.H. Cockayne, personal communication, 12 N. Bahic, R. Sch~iublin and K. J. Hemker, Philos. Mag. Lett., 64(1991)327. 13 N. Baluc and R. Sch~iublin,Weak beam image simulations of dislocations in ordered NiaAI, in preparation. 14 N. Baluc, J. Bonneville, K. J. Hemker, J. L. Martin, R. Sch~iublin and P. Spfitig, Mater. Sci. Eng., A164 (1993) 379.
373
A method for simulating electron microscope dislocation images R. Sch~iublin lnstitut de GOnie Atomique, Ecole Polytechnique F~d&ale de Lausanne, CH-IOI5 Lausanne (Switzerland)
P. Stadelmann Institut Interd~partemental de Microscopic Electronique, Ecole Polytechnique Fdd&ale de Lausanne, CH- I OI 5 Lausanne (Switzerland)
Abstract An image simulation program has been developed to quantitatively evaluate transmission electron microscopy (TEM) images of dislocation configurations. It takes into account many beams, includes linear anisotropic elasticity and uses the column approximation. The program simulates the image of 1 to 4 parallel dislocations and their associated planar faults under bright field, dark field and various weak-beam TEM conditions. Simulations of a variety of dislocations are presented here and it is shown that the use of a many-beam calculation is essential in the case of weak-beam identification of weakly dissociated dislocations.
1. Introduction
The resolution offered by weak-beam T E M allows for the direct observation of small lattice defects such as dissociated dislocations [1, 2]. This may yield a better understanding of the mechanical properties, especially of certain intermetallic compounds such as Ni3A1, which exhibits peculiar mechanical properties [3-5] related to dissociated dislocations [6]. Unfortunately results from the weak-beam method cannot be directly related to dislocation separations. The position of the peak does not correspond exactly to the actual dislocation position, and furthermore, there can be additional peaks under certain T E M conditions. Cockayne et al. proposed corrections on peak positions assuming linear isotropic elasticity [7]. This paper describes a new program of image simulation based on the work of Head et al. [8]. It is used to define reliable simulation conditions for two-beam and weak-beam observations. It is also shown that accurate dislocation distances can be measured under weakbeam conditions, by using this program to calculate dislocation image shifts. Head et al. calculated the images of dislocations using the Howie-Whelan equations, that take into account the transmitted beam and one diffracted beam only. The so-called generalized cross-section method was applied to reduce the number of integrations. This concept, based on the fact that the deformation field around a straight dislocation has a cylindrical symmetry, restricts the program to the calculation of straight, parallel and infinite dislocations. 0921-5093/93/$6.00
The geometry of the generalized cross-section defines a configuration where the dislocations intersect the sample surfaces and cannot be parallel to them. Nevertheless, it has been shown [8] that the displacement field of an infinite crystal can be successfully used to describe a dislocation in a thin foil when surface relaxations are ignored. The program of Head et al. has been extended to calculate images of 1 to 4 dislocations and their associated planar faults. The development of a linear differential equation system describing the propagation of the electrons in the sample was made. It allows simulation of images using more than two beams. It has been estimated [8] that about 90% of the total intensity was contained in the first two beams and it was deduced that the two-beam simulation was suitable for many situations. However, when using kinematic conditions, this may not be true any more and the need of a manybeam calculation may arise, especially in weak-beam conditions when the Ewald sphere intersects diffracted spots that are far from the transmitted spot. The intersected spot becomes excited and may contribute more strongly compared with diffracted spots which are closer to the transmitted spot. The number and type of beams that have to be taken into account in the calculation is therefore not obvious. They will be discussed in order to find the optimal simulation conditions, i.e. the simulation conditions where the number of beams is minimal to reduce the computing time together with a calculated image as alike as possible to real observation. © 1993 - Elsevier Sequoia. All fights reserved
374
R. Schiiublin, P. Stadelmann
/ Simulating electron microscope dislocation images
2. Theory To calculate the propagation of electrons in a faulted crystal, the dynamic theory of contrast (see Hirsch et al. [9]) is used. The crystal at a point r is described by a faulted potential V(r) that can be written as a Fourier series: V(r)=
[Ug exp(- 2:tigR(r))] exp(2~rigr)
beam
direction:/Z /z+g+Sg'~ 4g tO0/ ¢p,/
(1) Ewald sphere
-
The summation is done over all reciprocal lattice vectors g, with m e the electron mass and h the Planck constant. R(r) describes the displacement field around the lattice defect. The SchrSdinger's wave equation is: ~72tI/(r)+ ( ~ 2 me) [E + V(r)]W(I)=0
(2)
W(r) is the function associated with the electron wave that moves through the faulted crystal. The proposed solution to the SchrSdinger equation is: W ( r ) = 7 q~ exp(2xi(z +g+Sg)r)
The matrix M is symmetrical: A
A I
AI
BI
• ..
A i
0q~(r) = Z zri 0~/~ - h ~g-h qgh(r)exp[2zti(h- g)R(r) + 2:ri(Sh -- Sg)r]
(4)
The equations are to be integrated along the proper beam direction r/g, which is difficult to solve analytically. To simplify, we use the column approximation that allows integration along the same direction for all beams (6/6r/=d/dz). We can then write the equation system (4) in matrix form:
[Co(r)1 dq~(r)= Mq~(r)
where , ( r ) = [ qg'(r)[:
L q~£(r)j
(5)
...
A~"
C. Ai
Bi
Cii
".°
B.
An
The q0g(r)function is associated with the beams that come out of the sample, including the transmitted beam (beam 0) and the diffracted beams (beam 1 to n). The contrast intensity I recorded on the micrographs for a g image, is simply I = q0g(r).tpg*(r). Figure 1 shows the crystal and the beams in the reciprocal space. Z is the direction of the transmitted beam; g is the reciprocal lattice vector and sg the deviation parameter. (Z + g + Sg) is the direction of the beam with index g and we will take a co-ordinate r/g along this direction. When V(r) and W(r) are substituted in the Schr6dinger equation, a system of n differential equations of the first order with n unknowns qgg(r)is obtained:
dz
Fig. 1. Representation of a g(4g) weak beam.
(3)
g
;oYS2 ematic
~
1+ i/
A= Bi
¢0
(6)
f =L-
Cij = i~1
+'77-~i j
~,. defines the extinction distance which becomes large if beam i becomes far from the transmitted beam; ~'i is related to ~i through a material-dependent absorption constant. Equation (5) shows that the derivative of each beam is a linear combination of all the beams. The contribution of each beam in the linear combination is weighted by the matrix coefficients (6). The physical meaning of relation (5) is that the contribution of a beam i depends on the associated deviation parameter si (diffraction condition), the associated extinction distance ~i (material characteristics) and a cross-term ~i_j that has the expression of an extinction distance. High Igi] values correspond to large ~i and therefore beam gi has a weak contribution. Conversely, beam gi has a strong contribution when the deviation parameter sz is small. The importance of the contribution is a complex balance between the observation conditions s~ and the material characteristics (~). This has to be taken into account in the choice of the beams included in the
R. Schiiublin, P. Stadelmann
/
375
Simulating electron microscope dislocation images
calculation: they have to be close to the transmitted beam (small ~i) and with g~ situated inside and close to the Ewald sphere (smaller sg). The dependence on ~i / indicates that if we take for instance beam gg, then beams g~_~ and g~+~ will contribute strongly to beam gg as ~ i is small.
"cross-section"
thin foil with 1dislocation b=a[1, 1, 1] thickness : 155.52 nm
3. Simulations
In order to illustrate the capabilities of the program, different simulations were performed on dislocations in r-brass. This CuZn alloy exhibits strong anisotropy (A-- 8.5) and has a b.c.c, structure. The same material has been used by Head et al. [8] for various image simulations. We have compared our calculations with one of their examples by using the same conditions ( V ~ = 100 kV, C~ = 129, C12=110, C44=82 GPa). We then extended the calculations by including additional beams to the initial two beams. We consider successively the case of one dislocation (Fig. 2(a)) with g = [020] and two parallel dislocations (Fig. 2(b)) with g=[1, - 1 , 0]. In order to compare the contributions of the various beams, images of one dislocation have been calculated using an increasing number of beams. Figure 3 gives a series of simulated bright field images and two series of calculated weak beam images. Two different weakbeam conditions, g(3g) and g(4g), have been investigated. The image shift associated with neighboring dislocations can be corrected with comparisons to simulated images. Figure 4 presents calculated images of two dislocations which correspond to Fig. 2(b), with different distances of separation, and under dark-field and weak-beam conditions. The safest simulation conditions were chosen according to the results of Fig. 3. From the simulations it is obvious that the width of the image peaks is decreased by using weak-beam conditions and that the weak-beam condition is the most appropriate for observing dissociation distances.
4. Results and discussion
Figure 3(a) shows that in a bright field the image calculated with (0, g) does not change significantly through the addition of other beams, and indicates that the two-beam simulation conditions are suitable for bright field images. This simulated image is in satisfactory agreement with the one published by Head et al. [8]. These two points emphasize the reliability of the present program. As far as weak-beam images simulation are concerned, the first image of Fig. 3(b) and (c) has been calculated with five successive beams. We found that
b) foil ,norma
i i
i i
,
,.7;~.q'7.H:!..%~;.:i.;#~'s.~?~-~.',:ci.~:7.~7:~(~!+!cT,'~:'~.'~'U%'.'+~c."
• ÷:..: .....~..
~ .,:.,:e..~.
~
~ .-.:.:.~...:
......
:.~....:.~ .........
image
[1, 1, 3] thin foil with 2 dislocations in
(~,o, 1) bT,z=a[1, O,
1]
thickness : 55.3 nm image
Fig. 2. Simulation geometry: configurationwith (a) one dislocation, (b) two dislocations.
additional beams do not change the image significantly and are therefore considered as the most reliable ones. Simulations with less than five beams are presented in Fig. 3(b) and (c). For both weak-beam conditions, the image is not satisfactory when only the transmitted beam and the g beam are considered. For the g(3g) conditions, the image which corresponds to a four-beam calculation (0 to 3g) is identical to the reliable one. The image corresponding to three beams (0 to 2g) shows some similarities with the reliable one but with three other beams (0, g, 3g), it looks completely different. Therefore in this case, a rather reliable simulation requires at least three beams (0, g, 2g). For the g(4g) conditions, Fig. 3(c), the image which corresponds to a four-beam calculation (0-3g) is very reliable, while the others are not. The general rule that can be derived from these observations and, accordingly, to the above considerations on relation (5), is that the beams that have to be taken into account for the calculation are the following: the transmitted beam, the diffracted beam chosen to form the image and diffracted ones that are close to the Ewald sphere. It can be observed that the excited beam is not essential for the calculation. In what concerns dislocation separation measurements, Fig. 4(a) shows the dark field images are not appropriate for observing dislocation separation below 10 nm. Figure 4(b) suggests that the weak beam g(4g) is the most suitable condition for observing dislocations in fl-brass that are separated by distances close to
376
a) BF O(g)
R. Schiiublin, P. Stadelmann
/
Simulating electron microscope dislocation images
b) WB g(3g)
0 g
c) WB g(4g)
3g
0 to 4g
0 g 2g
0 g
0 g
0 g 2g 3g
0 g 2g
0 g 2g
-g to 3g
0 to 3g
-2g to 3g
0 g 3g
m m
m
0 to 3g
0 4g
Og 3g 4g
m
Fig. 3. Examples of simulated images of one dislocation. The beams used in the simulation are indicated on the left. (a) Bright field (two-beam case), (b) weak beam g(3g), (c) weak beam g(4g).
R. Schiiublin, P. Stadelmann
a) DF g(g)
/
Simulating electron microscope dislocation images
b) W B
377
c) WB g(4g)
d=5nm
2 nm
~g)
2 nm
5
g
5
nm
(2g)
nm
10
g
nm
(3g)
0 ~m
20 nm
74g )
40
40
nm
nm
1 I
Fig, 4. Examples of simulated images of two parallel dislocations. (a) Dark field (two-beam case): two beams (0, g). The separation distance is indicated on the left. (b) Weak beam: separation of 5 nm, six beams ( - g to 4g). The observation condition is indicated on the left. (c) g(4g) weak beam: six beams ( - g to 4g). The separation distance is indicated on the left.
5 nm, and Fig. 4(c) gives the calculated images for different dislocation distances. Figure 5 shows more quantitatively how the observed distance depends on the actual distance for a configuration close to Fig. 2(b) with b l = a [ 1 0 1 ] , b2=a[010], e - = [ - l 1 3 ] , g = [ l l 0 ] . For separations smaller than 10 TUn the correction is found to be important. It indicates that the image shift due to the displacement field overlap has to be taken into account in this case. The error introduced by the column approximation will only become important at
small distances [10]. Cockayne indicates a distance limit of about 0.5 nm in the g(4g) case [11].
5. Conclusion The development of a many-beam program to simulate T E M images of straight parallel dislocations in weak-beam conditions has been achieved.
378
R. Schiiublin, P. Stadelmann
/
Simulating electron microscope dislocation images
15
interest in this work. We also gratefully acknowledge the help of A. VtJser in calculating all the images appearing in this paper.
lO
References 5
o
5
lO
15
Fig. 5. Observed dislocations distance (nm) (simulation) as a function of the actual distance (nm). T h e use of many-beam calculations, as c o m p a r e d to two-beam calculations, appears to be necessary when imaging in weak-beam conditions, i.e. far from the Bragg condition. T h e program allows for the correction of the image shift through the comparison of T E M observed dislocation images and simulated images. Examples of this type of quantitative analysis are given in refs. 12 - 14.
Acknowledgments Support for this project was provided by Fonds National Suisse de la Recherche Scientifique. T h e authors would like to thank B. Viguier, Dr. N. Baluc, Dr. K. H e m k e r and Professor G. Vanderschaeve for fruitful discussions and Professor J. L. Martin for his
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