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The intersecting Kikuchi line technique; critical voltage at any voltage
Uhramicroscopy 17 (1985) 329-334 North-Holland, Amsterdam
329
T H E I N T E R S E C T I N G K I K U C H I L I N E T E C H N I Q U E ; C R I T I C A L V O L T A G E AT ANY V O L T A G E J. T A F T O Metallurgy and Materials Science Division, Brookhaven National Laboratoo', Upton, New York l ] 973, USA
and J. G J O N N E S Department of Physics', University of Oslo, Blindern, Oslo, Norwc~v Received 13 May 1985
The close relationship between the critical voltage and the intersecting Kikuchi line technique, two electron diffraction techniques for the accurate determination of structure factor values, is demonstrated by analytical treatment of simple diffraction conditions involving three beams. As an example the value of the 220 structure factor of cubic SiC is determined from convergent-beam electron diffraction using 100 keV electrons.
1. Introduction Among the techniques which have been developed to determine structure factor values from electron diffraction experiments, the Critical Voltage (CV) technique [1] is usually considered the most accurate, whereas the Intersecting Kikuchi Line (IKL) technique [2] may be more versatile. Both techniques are based on dynamical manybeam effects. Dynamical extinction of a reflection or Kikuchi line at a certain, usually several hundred keV, acceleration voltage is observed when the CV technique is used, and distances between intensity anomalies in the diffraction pattern are measured when using the I K L technique. Gjonnes and Hoier, in their original work on I K L [2], pointed out that these two techniques are closely related. The purpose of this paper is to demonstrate this relationship by comparing two simple three-beam situations. Finally, we determine the 220-structure factor of cubic SiC from convergent-beam diffraction patterns obtained by
using a conventional 100 keV transmission electron microscope.
2. Analytical treatment of critical voltage and intersecting Kikuchi lines Both the CV and IKL effects can be understood by considering three-beam diffraction conditions which can be solved analytically. Fig. la shows the diffraction condition using the CV technique. Here the reflection h = 2g is at the Bragg position. For simplicity we assume that the crystal has inversion symmetry along the g direction, which means that for appropriate choice of origin we have the following relationships between the Fourier potentials: U~=U~ and U/, = U/,. Fig. lb shows the simplest diffraction condition giving rise to intensity anomalies where Kikuchi lines intersect. We also here assume h to be at the Bragg position, and U~ = u~ and Uj, = Uj, which is fulfilled when the crystal has a mirror plane normal to h. The
0304-3991/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
J. Tafto, J. Gjonnes / Intersecting Kikuchi line technique
330
dispersion equations (see Howie [3] for notation and units) are thus identical for the two cases shown in fig. la and fig. lb, namely:
65,
-2 Vv V,,
- 2 Ky
/4
] <'1
U~
U~
- 2 K y + 2Ksv
L~
C j, = 0. Cv J
(1) Here the Fourier potential uni| cell
f, exp(2~rg • r )
vv= - .O1- () nl
t Vt'/?
(A et
(similar for Uj,). m and m o are the relativistic and rest mass of the incident electron, f, and r, are the atomic scattering amplitude and the coordinate in
the unit cell of atom i. K, y and s v are the wave vector, the Anpassung and the excitation error (all in A 1 units), and V (A~) is the volume of the crystal unit cell. A general discussion of three-beam electron diffraction, i.e., no symmetry assumptions, has been given by Gjonnes and Hoier [2]. Discussions of the special case of fig. l a in connection with the CV effect appear in ref. [4] and a detailed solution of eq. (1) in ref. [5]. Due to the symmetry of the problem the matrix (eq. (1)) can be reduced to one one-dimensional matrix and one two-dimensional matrix which are easily solved. The two Bloch waves associated with the two-dimensional matrix are referred to as symmetrical (S) Bloch waves because Cj, = Q , and the Bloch wave associated with the one-dimensional matrix is referred to as antisymmetrical (A) because Cj, = - C~ and C~ = 0. By inserting these relationships between the Bloch wave coefficients, C, we find (see ref. [5]): For the antisymmetrical (A) Bloch wave 2 K Y A = - - U h,
(2)
and for the symmetrical (S) Bloch waves (I) b= 2 g
0
2 K y ; 2 ' = ~ ( U h + 2 K s v +, ~__/ ( U h - 2 K s v ) - +
8
U2v
.
(3) Critical voltage, which means that the intensity of the reflection h = 2g vanishes, occurs when lal
2Ky~ z ' = 2KyA.
(4)
Notice that in fig. la the geometry is fixed and 2Ksv has the same value for all accelerating voltages. Thus, the only parameter that can be changed is the relativistic mass, and from eq. (4) we obtain critical voltage when rn
2 Ks.~ V h
",7 Q.
(bl Fig. 1. Three-beam diffraction conditions: (a) critical voltage, (b) intersecting Kikuchi lines.
h
(s)
where Vh and Vv are the Fourier potentials corresponding to the rest mass, m 0, Fig. 2a shows the Anpassungen 2K-{ as a function of the accelerating voltage for the case of fig. la, with cubic SiC as an example. Here g = 220 and h = 440. The scattering amplitudes of Si and C were taken from
J. Tafto, J. Gjonnes / Intersecting Ktkuchi hne technique
i
i
i
i
i
,.,/2K('1
a
0.6
i
i
i
i
331
i
b
~
0.I0 /
/ eJ~
g
).o5
/ g
----
0.4
2K ( I ) - 2 K (2);KS ;KS - min --
o<~
/
2K
//
.
/ ~: ~0.2
~ ~ : 0
/ - . . . . . . . . . .
O, h
~
.
.
.
.
.
.
---
/// ,/
J
0
O,h
/ --/ /
. . / ; /
-3.05 -
// ~ f ~ -I I
2K;KA "
DEGENERACY
'
I -0.2
I /
0
J,
2
3
4
5
6
7
8
--~
mo
-0.05
Ug2_Uh2 2 K s I= U ~
c)
35(~-2l
I /2Ks2=
Uh
0.05
2Ks T
Fig. 2. (a) Three-beam calculation of the Anpassung as a function of relativistic mass for SiC (220) reciprocal lattice row. CV occurs where 2 K T A and 2KT~ 21 intersect. The dotted lines g and 0, h are the kinematical dispersion surfaces, i.e.. /ale= L/,v = 0. (b) Three-beam calculation of the Anpassung as a function of 2Ksl35. Here h = 220 and g =135. 2K7~, and 2 Ky s.(2~ intersect at 2K,h. Minimum distance between 2KT~ 1) and 2Ky~ 21 at 2Ks2.
ref. [6]. The intersection 2K%~ = 2KT~ 2~ a p p e a r s for m / m o = 4.8, c o r r e s p o n d i n g to a critical voltage of nearly 2 meV, which is outside the range of most electron microscopes. In the I K L m e t h o d the voltage is kept c o n s t a n t a n d the excitation error, 2Ks~, is varied by rotating the crystal a r o u n d the h-direction. In practice this is achieved in the Kikuchi or the convergentb e a m diffraction pattern. The Kikuchi lines will a p p e a r at the position of the gaps at the dispersion surface, i.e., at 2 K y ~ - 2 K ' y ~ 2 ~ = m i n a n d at ,- I'h ^ Y s - = 2KyA, referring to eqs. (2) (4) or to the calculated section through the dispersion surface of SiC shown in fig. 2b. The calculation corres p o n d s to g = 335 and h = 220 in the configuration of fig. 1 b, i.e., s2> = 0.
3. Example from SiC Fig. 3 shows a Kikuchi p a t t e r n from cubic SiC taken in a J E M 100C at 100 kV. The line L at
$220 = 0 c o r r e s p o n d s to the section calculated in fig. 2b. N e a r the intersection with this line the two Kikuchi lines K~ (i35) and K , (315) are seen to be split into two h y p e r b o l a - l i k e segments. The contrast of the upper segment, which c o r r e s p o n d s to the d e g e n e r a c y in eq. (4) vanishes at the intersection with L. Here 2Ks,~-(U~, 2 - UI,)/CS,. The lower segment crosses L where 2KT~ I~- 2KT~ 2~ has m i n i m u m , which occurs when 2Ks1~5 = L;~_,,~ a c c o r d i n g to eq. (3). F o r a further discussion of the c o n t r a s t features, see ref. [2]. The geometry of the e x p e r i m e n t a l situation m a y be more easily unders t o o d by considering the c o r r e s p o n d i n g converg e n t - b e a m electron diffraction ( C B E D ) p a t t e r n which is shown schematically in fig. 4. We notice that the d i s t a n c e between the two intensity a n o m a l i e s is c~ - c~, where the center of the Laue circle, c, is the c o m p o n e n t of the K-vector in the p r o j e c t i o n under consideration, in this case the 554 projection. F o r a cubic structure the relationship between c and s v is
332
J. Tafto, J. @onnex / lnterse~lng Ktkuchi lit e lechnique
Fig. 3. 100 keV Kikuchi panern corresponding to fig. 2b.
2 K s = 2 c g - g2.
(6)
Easier than to measure c~ and c 2 is to measure c, - c I which is related to s by 2Ks~
-
2Ks 1 = b],
- ,
(U:
= 2(ce- c,)g.
""
(7)
in the present work we measured c ~ - c~ on a C B E D pattern which is expected to improve the accuracy as compared to a Kikuchi pattern. The C B E D pattern (fig. 5) was obtained from a wedge-shaped area of thickness as large as a couple of thousand A in order to sharpen the features
of interest. The C B E D condition which we found best suited for m e a s u r e m e n t was the one with the 315 and ]35 reflection rather than the ]35 and 220 near the Bragg position. These two diffraction conditions have the same symmetry, and the same reciprocal vectors are involved and give the same e, - cj. From fig. 5 we find "2 - c, - (0.0213 +_ 0.0005)(
2, 2, 5 ) / a ,
where a - 4.36 A is the cubic edge of the SiC unit cell. F r o m this m e a s u r e m e n t we get 2 K s 2 - 2Ks~ = 0.076 _+ 0.002 A 2. Based on Doyle and T u r n e r ' s scattering amplitudes for Si and C [6] we get U h - ~211=0-040'& 2 and U~= U 3 1 s = 0 . 0 1 4 A 2
J. Tafio, J. Gjonnes / Intersecting Kikuehi line technique
333
of higher-order reflections along the reciprocal lattice row have to be known with fairly high accuracy in order to determine U~, and thus many-beam dynamical calculations are necessary.
4. Conclusion
0
220
Fig. 4. S c h e m a t i c a l c o n v e r g e n t - b e a m d i f f r a c t i o n p a t t e r n c o r r e s p o n d i n g to fig. 2b a n d fig. 3. T h e small circles are the p o s i t i o n of i n c i d e n t a n d d i f f r a c t i o n b e a m s relative to the fixed c e n t e r of the L a u e circle e for t w o d i r e c t i o n s o f incidence. Small filled circles with c = c I give s/~ 5 = s I < 0 a n d small o p e n circles with c e 2 give sl~ s = s 2 > 0 .
at 100 keV, and the theoretical value is thus (eq. (7)): 2 K s 2 - 2 K s 1 = 0.075 ,~ 2. Using the approximation 2 K s t = - UI, which applies when Uh >> U~ would lead to the experimental value Uz20 = 0.038 ,~ 2 rather than 0.040 A 2 showing that a reasonable accuracy may be obtained without knowing the value of other structure factors [2]. This is in contrast to the CV technique where U2.~ and also the structure factor
By analytical treatment of simple three-beam electron diffraction the relationship between the CV and the I K L effect has been illustrated. We have also shown that for light element crystals analytical three-beam calculation may give good agreement when the I K L technique is used. However, for more accurate measurements, which can be obtained by using energy-filtered CBED patterns that greatly improve the contrast [7], manybeam computer calculations are necessary to fully utilize the experimental data. Whether energyfiltered 100 keV CBED patterns of I K L features will give as high an accuracy as the CV technique [8] and conventional I K L patterns at high voltages [9] remains to be seen. Finally, we point out that apart from the possibility of focusing an electron beam on a small crystal area, electron diffraction has another attractive property, namely that low-index reflections are very sensitive to the chemical state of the atoms, e.g., valency state and charge transfer. Most sensitive, and also most interesting, are relatively weak first-order reflections into which the different atoms scatter out of phase [10]. In such cases V2~ is often larger than V~ and it follows from eq. (5) that m / m o is negative, which means that the CV technique cannot be used with electrons.
334
J. Tajto, J. Glotme.s / lnter~ectmV Kikuchi lille technique
Fig. 5. CBED pattern showing the direct beam disk when the 135 and the 315 reflection are at the Bragg position.
Acknowledgement Support from US Department of Energy, Division of Materials Sciences, Office of Basic Energy Sciences under Contract No. DE-AC0276CH00016 is gratefully acknowledged.
References [1] D. Watanabe, R. Uyeda and M. Kogiso, Acta Cryst. A24 (1968) 249. [2] J. Gjonnes and R. Hoier, Acta Cryst. A27 (1971) 313. [3] A. Howie, in: Modern Diffraction and Imaging Techniques in Materials Science, Eds. S. Amelinckx, R. Gevers,
[4] [5] [6]
[7]
[8] [9] [10]
G. Remaut and J. Van Landuyt (North-Holland. Amsterdam, 1970) pp. 295-339. J.S. Lally, C.J. Humphreys, A.J.F. Methere[I and R.M. Fisher, Phil. Mag. 25 (1972) 321. L. Reimer, Transmission Electron Microscopy (Springer, Berlin, 1984) pp. 302 306. International Tables for X-Ray Crystallography, Vol. 4, Eds. J. Ibers and W. Hamilton (Kynoch, Birmingham, 1974) p. 152. A. Higgs and O.L. Krivanek, in: Proc. 39th Annual EMSA Meeting, Atlanta, GA, 1981, Ed. G.W. Bailey (Claitor's, Baton Rouge, LA, 1981) p. 346. I). Watanabe, R. Uyeda and A. Fukuhara, Acta Cryst. A24 (1968) 580. O. Terasaki, D. Watanabe and J. Gjonnes, Acta Cryst. A35 (1979) 895. K. l s h i z u k a a n d J . Tafto, ActaCryst. B40(1984) 332.
329
T H E I N T E R S E C T I N G K I K U C H I L I N E T E C H N I Q U E ; C R I T I C A L V O L T A G E AT ANY V O L T A G E J. T A F T O Metallurgy and Materials Science Division, Brookhaven National Laboratoo', Upton, New York l ] 973, USA
and J. G J O N N E S Department of Physics', University of Oslo, Blindern, Oslo, Norwc~v Received 13 May 1985
The close relationship between the critical voltage and the intersecting Kikuchi line technique, two electron diffraction techniques for the accurate determination of structure factor values, is demonstrated by analytical treatment of simple diffraction conditions involving three beams. As an example the value of the 220 structure factor of cubic SiC is determined from convergent-beam electron diffraction using 100 keV electrons.
1. Introduction Among the techniques which have been developed to determine structure factor values from electron diffraction experiments, the Critical Voltage (CV) technique [1] is usually considered the most accurate, whereas the Intersecting Kikuchi Line (IKL) technique [2] may be more versatile. Both techniques are based on dynamical manybeam effects. Dynamical extinction of a reflection or Kikuchi line at a certain, usually several hundred keV, acceleration voltage is observed when the CV technique is used, and distances between intensity anomalies in the diffraction pattern are measured when using the I K L technique. Gjonnes and Hoier, in their original work on I K L [2], pointed out that these two techniques are closely related. The purpose of this paper is to demonstrate this relationship by comparing two simple three-beam situations. Finally, we determine the 220-structure factor of cubic SiC from convergent-beam diffraction patterns obtained by
using a conventional 100 keV transmission electron microscope.
2. Analytical treatment of critical voltage and intersecting Kikuchi lines Both the CV and IKL effects can be understood by considering three-beam diffraction conditions which can be solved analytically. Fig. la shows the diffraction condition using the CV technique. Here the reflection h = 2g is at the Bragg position. For simplicity we assume that the crystal has inversion symmetry along the g direction, which means that for appropriate choice of origin we have the following relationships between the Fourier potentials: U~=U~ and U/, = U/,. Fig. lb shows the simplest diffraction condition giving rise to intensity anomalies where Kikuchi lines intersect. We also here assume h to be at the Bragg position, and U~ = u~ and Uj, = Uj, which is fulfilled when the crystal has a mirror plane normal to h. The
0304-3991/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
J. Tafto, J. Gjonnes / Intersecting Kikuchi line technique
330
dispersion equations (see Howie [3] for notation and units) are thus identical for the two cases shown in fig. la and fig. lb, namely:
65,
-2 Vv V,,
- 2 Ky
/4
] <'1
U~
U~
- 2 K y + 2Ksv
L~
C j, = 0. Cv J
(1) Here the Fourier potential uni| cell
f, exp(2~rg • r )
vv= - .O1- () nl
t Vt'/?
(A et
(similar for Uj,). m and m o are the relativistic and rest mass of the incident electron, f, and r, are the atomic scattering amplitude and the coordinate in
the unit cell of atom i. K, y and s v are the wave vector, the Anpassung and the excitation error (all in A 1 units), and V (A~) is the volume of the crystal unit cell. A general discussion of three-beam electron diffraction, i.e., no symmetry assumptions, has been given by Gjonnes and Hoier [2]. Discussions of the special case of fig. l a in connection with the CV effect appear in ref. [4] and a detailed solution of eq. (1) in ref. [5]. Due to the symmetry of the problem the matrix (eq. (1)) can be reduced to one one-dimensional matrix and one two-dimensional matrix which are easily solved. The two Bloch waves associated with the two-dimensional matrix are referred to as symmetrical (S) Bloch waves because Cj, = Q , and the Bloch wave associated with the one-dimensional matrix is referred to as antisymmetrical (A) because Cj, = - C~ and C~ = 0. By inserting these relationships between the Bloch wave coefficients, C, we find (see ref. [5]): For the antisymmetrical (A) Bloch wave 2 K Y A = - - U h,
(2)
and for the symmetrical (S) Bloch waves (I) b= 2 g
0
2 K y ; 2 ' = ~ ( U h + 2 K s v +, ~__/ ( U h - 2 K s v ) - +
8
U2v
.
(3) Critical voltage, which means that the intensity of the reflection h = 2g vanishes, occurs when lal
2Ky~ z ' = 2KyA.
(4)
Notice that in fig. la the geometry is fixed and 2Ksv has the same value for all accelerating voltages. Thus, the only parameter that can be changed is the relativistic mass, and from eq. (4) we obtain critical voltage when rn
2 Ks.~ V h
",7 Q.
(bl Fig. 1. Three-beam diffraction conditions: (a) critical voltage, (b) intersecting Kikuchi lines.
h
(s)
where Vh and Vv are the Fourier potentials corresponding to the rest mass, m 0, Fig. 2a shows the Anpassungen 2K-{ as a function of the accelerating voltage for the case of fig. la, with cubic SiC as an example. Here g = 220 and h = 440. The scattering amplitudes of Si and C were taken from
J. Tafto, J. Gjonnes / Intersecting Ktkuchi hne technique
i
i
i
i
i
,.,/2K('1
a
0.6
i
i
i
i
331
i
b
~
0.I0 /
/ eJ~
g
).o5
/ g
----
0.4
2K ( I ) - 2 K (2);KS ;KS - min --
o<~
/
2K
//
.
/ ~: ~0.2
~ ~ : 0
/ - . . . . . . . . . .
O, h
~
.
.
.
.
.
.
---
/// ,/
J
0
O,h
/ --/ /
. . / ; /
-3.05 -
// ~ f ~ -I I
2K;KA "
DEGENERACY
'
I -0.2
I /
0
J,
2
3
4
5
6
7
8
--~
mo
-0.05
Ug2_Uh2 2 K s I= U ~
c)
35(~-2l
I /2Ks2=
Uh
0.05
2Ks T
Fig. 2. (a) Three-beam calculation of the Anpassung as a function of relativistic mass for SiC (220) reciprocal lattice row. CV occurs where 2 K T A and 2KT~ 21 intersect. The dotted lines g and 0, h are the kinematical dispersion surfaces, i.e.. /ale= L/,v = 0. (b) Three-beam calculation of the Anpassung as a function of 2Ksl35. Here h = 220 and g =135. 2K7~, and 2 Ky s.(2~ intersect at 2K,h. Minimum distance between 2KT~ 1) and 2Ky~ 21 at 2Ks2.
ref. [6]. The intersection 2K%~ = 2KT~ 2~ a p p e a r s for m / m o = 4.8, c o r r e s p o n d i n g to a critical voltage of nearly 2 meV, which is outside the range of most electron microscopes. In the I K L m e t h o d the voltage is kept c o n s t a n t a n d the excitation error, 2Ks~, is varied by rotating the crystal a r o u n d the h-direction. In practice this is achieved in the Kikuchi or the convergentb e a m diffraction pattern. The Kikuchi lines will a p p e a r at the position of the gaps at the dispersion surface, i.e., at 2 K y ~ - 2 K ' y ~ 2 ~ = m i n a n d at ,- I'h ^ Y s - = 2KyA, referring to eqs. (2) (4) or to the calculated section through the dispersion surface of SiC shown in fig. 2b. The calculation corres p o n d s to g = 335 and h = 220 in the configuration of fig. 1 b, i.e., s2> = 0.
3. Example from SiC Fig. 3 shows a Kikuchi p a t t e r n from cubic SiC taken in a J E M 100C at 100 kV. The line L at
$220 = 0 c o r r e s p o n d s to the section calculated in fig. 2b. N e a r the intersection with this line the two Kikuchi lines K~ (i35) and K , (315) are seen to be split into two h y p e r b o l a - l i k e segments. The contrast of the upper segment, which c o r r e s p o n d s to the d e g e n e r a c y in eq. (4) vanishes at the intersection with L. Here 2Ks,~-(U~, 2 - UI,)/CS,. The lower segment crosses L where 2KT~ I~- 2KT~ 2~ has m i n i m u m , which occurs when 2Ks1~5 = L;~_,,~ a c c o r d i n g to eq. (3). F o r a further discussion of the c o n t r a s t features, see ref. [2]. The geometry of the e x p e r i m e n t a l situation m a y be more easily unders t o o d by considering the c o r r e s p o n d i n g converg e n t - b e a m electron diffraction ( C B E D ) p a t t e r n which is shown schematically in fig. 4. We notice that the d i s t a n c e between the two intensity a n o m a l i e s is c~ - c~, where the center of the Laue circle, c, is the c o m p o n e n t of the K-vector in the p r o j e c t i o n under consideration, in this case the 554 projection. F o r a cubic structure the relationship between c and s v is
332
J. Tafto, J. @onnex / lnterse~lng Ktkuchi lit e lechnique
Fig. 3. 100 keV Kikuchi panern corresponding to fig. 2b.
2 K s = 2 c g - g2.
(6)
Easier than to measure c~ and c 2 is to measure c, - c I which is related to s by 2Ks~
-
2Ks 1 = b],
- ,
(U:
= 2(ce- c,)g.
""
(7)
in the present work we measured c ~ - c~ on a C B E D pattern which is expected to improve the accuracy as compared to a Kikuchi pattern. The C B E D pattern (fig. 5) was obtained from a wedge-shaped area of thickness as large as a couple of thousand A in order to sharpen the features
of interest. The C B E D condition which we found best suited for m e a s u r e m e n t was the one with the 315 and ]35 reflection rather than the ]35 and 220 near the Bragg position. These two diffraction conditions have the same symmetry, and the same reciprocal vectors are involved and give the same e, - cj. From fig. 5 we find "2 - c, - (0.0213 +_ 0.0005)(
2, 2, 5 ) / a ,
where a - 4.36 A is the cubic edge of the SiC unit cell. F r o m this m e a s u r e m e n t we get 2 K s 2 - 2Ks~ = 0.076 _+ 0.002 A 2. Based on Doyle and T u r n e r ' s scattering amplitudes for Si and C [6] we get U h - ~211=0-040'& 2 and U~= U 3 1 s = 0 . 0 1 4 A 2
J. Tafio, J. Gjonnes / Intersecting Kikuehi line technique
333
of higher-order reflections along the reciprocal lattice row have to be known with fairly high accuracy in order to determine U~, and thus many-beam dynamical calculations are necessary.
4. Conclusion
0
220
Fig. 4. S c h e m a t i c a l c o n v e r g e n t - b e a m d i f f r a c t i o n p a t t e r n c o r r e s p o n d i n g to fig. 2b a n d fig. 3. T h e small circles are the p o s i t i o n of i n c i d e n t a n d d i f f r a c t i o n b e a m s relative to the fixed c e n t e r of the L a u e circle e for t w o d i r e c t i o n s o f incidence. Small filled circles with c = c I give s/~ 5 = s I < 0 a n d small o p e n circles with c e 2 give sl~ s = s 2 > 0 .
at 100 keV, and the theoretical value is thus (eq. (7)): 2 K s 2 - 2 K s 1 = 0.075 ,~ 2. Using the approximation 2 K s t = - UI, which applies when Uh >> U~ would lead to the experimental value Uz20 = 0.038 ,~ 2 rather than 0.040 A 2 showing that a reasonable accuracy may be obtained without knowing the value of other structure factors [2]. This is in contrast to the CV technique where U2.~ and also the structure factor
By analytical treatment of simple three-beam electron diffraction the relationship between the CV and the I K L effect has been illustrated. We have also shown that for light element crystals analytical three-beam calculation may give good agreement when the I K L technique is used. However, for more accurate measurements, which can be obtained by using energy-filtered CBED patterns that greatly improve the contrast [7], manybeam computer calculations are necessary to fully utilize the experimental data. Whether energyfiltered 100 keV CBED patterns of I K L features will give as high an accuracy as the CV technique [8] and conventional I K L patterns at high voltages [9] remains to be seen. Finally, we point out that apart from the possibility of focusing an electron beam on a small crystal area, electron diffraction has another attractive property, namely that low-index reflections are very sensitive to the chemical state of the atoms, e.g., valency state and charge transfer. Most sensitive, and also most interesting, are relatively weak first-order reflections into which the different atoms scatter out of phase [10]. In such cases V2~ is often larger than V~ and it follows from eq. (5) that m / m o is negative, which means that the CV technique cannot be used with electrons.
334
J. Tajto, J. Glotme.s / lnter~ectmV Kikuchi lille technique
Fig. 5. CBED pattern showing the direct beam disk when the 135 and the 315 reflection are at the Bragg position.
Acknowledgement Support from US Department of Energy, Division of Materials Sciences, Office of Basic Energy Sciences under Contract No. DE-AC0276CH00016 is gratefully acknowledged.
References [1] D. Watanabe, R. Uyeda and M. Kogiso, Acta Cryst. A24 (1968) 249. [2] J. Gjonnes and R. Hoier, Acta Cryst. A27 (1971) 313. [3] A. Howie, in: Modern Diffraction and Imaging Techniques in Materials Science, Eds. S. Amelinckx, R. Gevers,
[4] [5] [6]
[7]
[8] [9] [10]
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