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A method for determining foil thicknesses in TEM by using convergent beam electron diffraction under weak beam conditions
Micron,Vol. 26, No. 6, pp. 539-543, 1995 Copyright© 1996ElsevierScienceLtd Printed in Great Britain. All rights reserved 0968--4328/96$9.50+ 0.00
Pergamon 0968-4328(95)00008-X
A Method for Determining Foil Thicknesses in TEM by Using Convergent Beam Electron Diffraction Under Weak Beam Conditions K. Z. B O T R O S
Fuel Channel Components Branch, Reactor Materials Division, Atomic Energy of Canada Ltd, Chalk River Laboratories, Station 55, Chalk River, Ontario, Canada KOJ 1JO (Received 14 November 1995)
Al~traet--In the present work, convergent beam electron diffraction was studied in zirconium (a material of intermediate atomic number) at 300 keV, under weak beam diffraction conditions. For a particular thickness, the details in an observed low order disc were matched to those calculated using the multibeam dynamical theory. This presents the possibility of determining foil thickness over a wide range, with an estimated experimental accuracy of ~ 7"/'0or less. In contrast to other convergent beam techniques, the present method, which uses weak beam conditions, can employ commonly-occurringlow order reflections to extract thicknesses. A simple equation based on the two beam approximation, is derived to determine foil thickness (to within ~10%) without resorting to detailed image matching. This equation can be used for a rough estimate of foil thickness while carrying out TEM observations. Copyright © 1996 Elsevier Science Ltd. Key words: foil thickness determination, convergent beam electron diffraction, weak beam conditions.
I. I N T R O D U C T I O N Foil thickness is an important parameter in m a n y analytical electron microscope (AEM) investigations, e.g. in assessing the density and nature of dislocations, in obtaining accurate micro-chemical analyses, and in studies of radiation-induced defects. Small changes in foil thickness can strongly affect defect contrast (see for example Hertel et al., 1977); and therefore accurate thickness determination is essential in correct defect identification (see, for example, H e a d et al., 1973). Kelly et al. (1975) have developed a method to determine foil thickness. This method employs convergent beam electron diffraction (CBED) when a low order reflection, g, satisfies its Bragg condition. The pattern in the g-disc consists of fringes which are normally symmetrical a b o u t the center of the disc, and foil thickness can be extracted f r o m the positions of such fringes. A criterion was outlined by Kelly et al. (1975) for the order of the reflection to be used. These authors warned of using some low order reflections, because systematic reflections can give rise to a complex C B E D pattern which m a k e thickness determination difficult. It is a c o m m o n practice to operate a modern A E M at its highest electron energy (200-400 keV) to take advantage of the improved resolution and greater penetration. At such electron energies, and especially in materials of intermediate atomic number, the effects of systematic reflections are expected to be quite strong. Consequently, problems can arise in extracting foil thickness, if the method proposed by Kelly et al. (1975) is used. A possible solution is to employ progressively higher order reflec-
tions (Cann, 1978). However, high order reflections lead to larger extinction distances which narrows the range of foil thicknesses that can be determined. It is possible to avoid the difficulties with strong effects of systematic reflections by choosing weak beam diffraction conditions (Cockayne et al., 1969), because diffraction is essentially kinematic under such conditions. The present work explored this possibility by studying the nature of the C B E D pattern in a low order disc obtained at 300 keV, under weak b e a m diffraction conditions. F r o m the pattern in such discs, extracting the foil thickness was explored in a material of intermediate atomic number, namely hcp zirconium. The nature of the C B E D pattern in the g-disc, resulting under weak b e a m conditions, is presented in section 2, and experimental examples are given in section 4.
H. N A T U R E OF T H E CBED P A T T E R N I N A G-DISC O B T A I N E D U N D E R W E A K B E A M DIFFRACTION CONDITIONS F o r a given incident b e a m direction, the amplitude of a diffracted beam ~bs (t), at the b o t t o m surface of a crystal of thickness t, can be written using the Bloch wave formulation (Hirsch et al., 1965) of the dynamical theory of electron diffraction as, N ~)g(t) : ~ e(0k)e(k) exp(27~i?(k)t) k=l 539
(1)
540
K.Z. Botros
where N in Eqn (1) is the number of systematic reflections considered, and it is also the number of Bloch waves which are excited. Assuming all Bitch waves to propagate along the zdirection, the flh Bitch wave will have the form b(J) = E
C(gj) exp(2qbi?(J)t)
(2)
g
In Eqn (2), cg03 is the b o l d {g}th element of the Bloch wave, and ~(/)= (kz0) - K z ) , where k~) is the z-component of the Bloch wave vector, Kz is the z-component of the inddent electron wave vector after correction due to the mean inner crystal potential. Under weak beam diffraction conditions, there are mainly (Cockayne, 1972) two Bloch waves (i and j ) contributing to the diffracted beam amplitude. Under such conditions, Bitch wave symmetry necessitates (Cockayne, 1972) that: C i)s'(i) O~g :
--°(J)e(J) = C "0 g
(3)
Substituting Eqn (3) in Eqn (1) gives fl~g(t) ~
cexp(2ni~(i)t)(1 - exp(2ni(? O) - ?(i))t))
(4)
The extinction distance under weak beam conditions effectively (Cockayne, 1972) given by ~g ~ (7 (]) --y(O) -1, and when this is substituted into equation (4), the intensity of the diffracted beam Ig, will be
is
Ig(t) ~ c2(1 - cos(2rct/~g))
(5)
Substituting for the extinction distance, ~g, its approximate value ~ US e under weak beam diffraction conditions (see Cockayne et al., 1969; Cockayne, 1972) gives
Ig( t) ~-. c2(1 -- cos(2nSgt) )
(6)
The condition for a minimum in the diffracted beam intensity is when (Set) is an integer, and for a maximum when (Set) has a value of n/2 (where n is an odd integer). Thus Eqn (6) predicts the pattern in the g-disc to consist of a number of parallel fringes with intensity alternating from bright to dark (maxima and minima). Equation (6) also predicts that the g-disc will exhibit high fringe contrast, because the maximum intensity is ~2e 2 while minimum intensity is almost zero. There will be a spread ASg in the values of the deviation parameter across a CBED disc from a minimum value Stain, to a maximum value of Smax- If two dark fringes occur at the edges of the disc, the first will occur when (tSmin) is a given integer, and the last is determined by (tSmax) being another integer. The number of dark fringes, m, in the g-disc will then relate Stain, Sm,x to the foil thickness t, through a simple relationship: t = m(A Sg) -1
(7)
In a general case dark fringes might not occur at the edges of the disc, i.e. at S~n or Sm~,. In such a situation,
a fraction should be added to the integral number of dark fringes counted in the disc. Thus, m, in Eqn (7) may become a real number to take into account a bright fringe which might appear at the edge of the disc. Equation (7) suggests that for a foil of thickness t, the g-disc will contain a number of fringes which only depend upon the difference (S,~x-Stain). If the width ASg of the CBED disc is measured and the number of fringes in it are appropriately counted, then Eqn (7) can be used to obtain the foil thickness.
m . MATERIALS AND M E T H O D S Two types of zirconium foils were employed in the present study. The first was prepared from material of ultra high purity, and the other from a Zr-2.5Nb pressure tube material used in C A N D U (CANadian Deuterium Uranium) reactors. All the CBED observations were carried out using a Philips CM30 electron microscope fitted with a twin objective lens, a double tilt holder, and operated at 300keV. Initially, a small second condenser aperture (usually 30mm) was used, and for a particular sample area chosen, tilting was carried out so that mainly one row of systematic reflections could be seen on the microscope screen. The sample was further tilted, so that the lowest order reflection in the systematic row had its deviation parameter, Sg, corresponding to weak beam diffraction (i.e. Sg>10-1nm -1 as defined by Cockayne, 1972). Caution was exercised to ensure that no low order nonsystematic reflections were close to their Bragg condition. The incident electron probe was then made convergent. The range of convergence angles 20is was 5 x l 0 - 3 - 1 2 x l 0 - 3 r a d . This was achieved by using different sizes of the second condenser aperture. The resulting CBED pattern was recorded on a photographic plate using an appropriately large camera length for clarity. The angular spread in a recorded CBED pattern depends on the beam convergence angle, 2~s, which gives rise to a spread ASg in the deviation parameter Sg, spanning values from a maximum Sm~x, to a minimum Stain. The center of the g-disc will be at a deviation Sg=(Smm+ASg/2). The spread ASs, depends on the Bragg angle of the reflection employed. A range of 'widths' for the g-disc were explored (see Table 1 for typical values for some commonly occurring reflections in Zr). The central value of Sg (corresponding to the middle of the disc) was determined from the position of the Kikuchi lines. Under weak beam diffraction conditions these lines Table 1. Typical range of 'width' in the convergent beam disc explored for different reflections in Zr (plane spacing are given). Disc width is expressed in fractions of 08, the corresponding Bragg angle Reflection
d (nm)
Range of disc width explored
0002 11-20
0.258 0.162
~1.4 00oo2to ~1.8 0ooo2 ~1.2 011-2o to ,-d.6 011-2o
Foil Thickness in TEM
were located with higher accuracy than is possible when using strong beam diffraction conditions, as required by Kelly et al., 1975). The spread in 6Sg in a recorded g-disc was determined from its diameter and its distance to the 0disc (after suitable calibration of angular spread). Accuracy in determining Sg, and 6Sg were estimated to be ~ 2 % , compared with ~ 5 % reported by Cann (1978), under strong beam diffraction conditions. The values of 6Sg and Sg were used to calculate the intensity of a particular diffracted beam against angular deviation. Intensity was converted to 256 gray levels, thus generating a theoretical CBED pattern for a more meaningful comparison to experimental observations. The CBED-patterns calculated by assuming different thicknesses showed sensitivity to small variations in foil thickness. An example of this is shown in Fig. la,b where visible changes in the CBED pattern (e.g. in the number of dark fringes) occur due to a small foil thickness variation. Thus to determine a foil thickness in the present work, thickness was varied in the calculation until good agreement was obtained between theory and experimental CBED patterns. The present calculations were based on the scattering matrix approach of the dynamical theory of electron diffraction (see Hirsch et al., 1965), which assumes the column approximation. The effects of absorption were taken into account by assuming the ratio of the imaginary to real part of the lattice potential to be 0.1. The value of this ratio was found to be not critical, as calculations showed that ratios between 0.06 and 0.12 gave no discernible effect on the simulated CBED patterns.
541
Fig. 2. The grain structure in a longitudinal section (i.e. foil normal is along the length of the tube) of Zr-2.5 Nb pressure tube. Most of the HCP grains in such material (such as those marked 'A') give diffraction patterns which contain the (0002) reflection.
in a g-disc when weak beam diffraction conditions are chosen. In this section, examples of observations in Zr2.5Nb pressure tube samples are presented. The agreement between observations and results of theoretical calculations are also presented. HCP ct-Zr grains in pressure tube material exhibit strong texture. For example, in a longitudinal section
(a)
IV. RESULTS OF OBSERVATIONS AND THICKNESS DETERMINATION. Arguments in Section 2 point to the possibility that foil thickness can be determined from the CBED pattern
(a)
(b)
.
(b)
Fig. 1. Simulated multibeam CBED patterns in Zr assuming the (11-20) reflection satisfying weak beam diffraction conditions at 300 keV. (Sm~-Sn~n) was assumed 1.6ql 1-2o. The foil thickness assumed in (a) was 95rim, and for Co) 99nm. Such a small change in thickness can be seen to give rise to visible changes in the CBED pattern. The resulting dark fringes are numbered.
~'~
:: ~
~
~.
O000-disc
O002-disc
Fig. 3. (a) Bright field and dark field CBED discs observed in a foil region from a longitudinal section of Zr-2.5Nb pressure tube. Observations were carried out at 300 keV and used the (0002) reflection under weak beam diffraction. Each disc had an angular width ~ 1.50o0o2. The deviation parameter at the center of the 0002-disc was S00o2~1.03 x l0 - l nm -1. Co) Simulated multibeam CBED discs calculated under the same conditions leading to the convergent beam pattern in (a). The foil t h i c k n e s s assumed was 240 nm. Note the weak contrast in the bright field disc.
542
K . Z . Botros
(with a normal along the length of the pressure tube), the majority of =-grains (some are marked as 'A' in Fig. 2) have normals near the (2-1-10) and (01-10) directions. The diffraction patterns of these foils frequently contain the low order (0002) reflection. It was then important to explore the use of this reflection to obtain a CBED pattern under weak beam conditions. An example is shown in Fig. 3a. In this figure bright field and dark field discs are shown. While such pattern was being observed on the microscope's screen, the bright field 0000-disc exhibited poor contrast and was of much higher intensity than the dark field 0002-disc which exhibited high contrast. The exposure time required to record the details of the dark field disc on the photographic plate, gave an overexposed bright field disc (see Fig. 3a). In Fig. 3b, results of calculations are presented. These calculations assumed the experimental diffraction conditions in obtaining Fig. 3a, and took into account the effects of 15 systematic reflections. The bright field disc in Fig. 3b shows the poor contrast that was experimentally observed, and because of this, no calculated bright field discs will be subsequently presented. On the other hand, many observed 0002discs from different foil regions were compared to theory. In each case good matching could be obtained between calculated and experimental 0002-discs, for an
appropriate foil thickness. An example of this can be seen in Fig. 3a,b. Weak beam observations were also carried out for other reflections in Zr. In Fig. 4a, an experimental example is shown for the (11-20) reflection. Fig. 4b shows the result of the corresponding multibeam calculation. Again, excellent agreement between observation and results of theoretical multibeam calculations can be seen by comparing Fig. 4a to Fig. 4b. This type of agreement was obtained between theory and experiment for a variety of observations covering a wide range of foil thicknesses (~ 35-~300 rim). The two beam theory could also be used to match experimental observations. The predicted thickness was, however, slightly different from that obtained from multibeam calculations. For example, good agreement between simulated two beam calculations and the experimental (0002) disc shown in Fig. 3a was obtained when foil thickness was assumed to be ~250 rim. This is only ~ 4 % different from the thickness predicted by multibeam calculations. Smaller differences were seen in other cases involving higher order reflections. The 4% difference found for the (0002) case, was nearly equal to that between the two beam and the multibeam extinction distances calculated under weak beam conditions. Finding that good agreement can exist between ob-
(a)
(b)
1120-disc Fig. 4. (a) C B E D diffraction pattern in a Z r - 2 . 5 N b foil obtained at 300keV using the (11-20) reflection under weak b e a m diffraction conditions. The angular width o f each o f the observed bright, dark field discs was ~ 1.5011_20. The deviation parameter at center o f the 11-20-disc was S]1_2o~2.6 x 10 - I n m - ] , a n d ( S m ~ - S m j , ) ~ 0 . 5 6 x 10 -~ n m - I . Co) Simulated multibeam 11-20disc, calculated under the same conditions leading to the convergent b e a m pattern in (a). The foil thickness a s s u m e d was 226 urn.
Foil Thickness in TEM servations and two beam calculations illustrates that under weak beam conditions, contrast in CBED pattern arises mainly from the interaction of two Bloch waves, as indicated in Section 2. Equation (7) in Section 2 offered the possibility of thickness determination without resorting to image matching. This is illustrated by an example for the (11-20) case in Fig. 4a. The l l-20-disc has a width (Smax--Smin) ~ 0 . 5 6 x 1 0 - 1 n m -1, and can be seen to contain 12.5 dark fringes. The thickness calculated from Eqn (7) is ~ 2 2 3 n m , as compared with ~ 2 2 6 n m obtained by a detailed image matching procedure. Such agreement indicates the possibility of using Eqn (7) for foil thickness determination.
V. D I S C U S S I O N Foils in the thickness range 30°300 nm are useful in studies of lattice defects in a material like Zr and Z r 2.5Nb at intermediate electron energies. The present work shows that specimen thickness can be determined when the experimental CBED pattern obtained under weak beam conditions is in good agreement with details o f a simulated disc. This method, discussed in Sections 2 to 5, offers the following possibilities: (a) Permits defect observations and thickness determinations over a wide range while using the microscope at intermediate incident electron energies (200-400 keV); (b) Easy application in foils of highly textured materials. The method can employ commonly occurring reflections, eliminating the need for complicated tilting procedures searching for suitable reflections; (c) Estimating thickness during A E M observations without resorting to detailed image matching procedures, can be done by counting the number of fringes in the CBED disc and using Eqn (7). There are limitations in using the method of Kelly et al. (1975) at intermediate electron energies for materials o f intermediate atomic number. At 200keV, Cann (1978) showed that the (11-20) is the lowest order reflection to employ in Zr to determine foil thickness. At 300 keV, Botros and Cann (1995) showed that if the (1120) reflection is chosen, the correct thickness determination requires the multibeam extinction distance, and that such determination is limited to a narrow range of ~1500250nm. In contrast to this, the present method offers the possibility of determination over a wider range, 30-300 nm. An important condition in employing the present method, is the choice of weak beam orientations such that no low order non-systematic reflection is close to its Bragg condition. This may be difficult to achieve in foil regions which are excessively thin and bent. In addition,
543
when accelerating voltages in excess of ,-~500 keV are used, difficulty in employing the present method may arise, at least in utilizing low order reflections like (0002) in Zr. This may be due to problems in realizing good weak beam conditions and in avoiding effects of nonsystematic reflections. Finally, it is important to comment on the experimental uncertainty of thickness determination expected from different convergent beam methods. Kelly et al. (1975) gave an optimistic ,-~2% in copper, while Cann (1978) estimated experimental uncertainty in zirconium to be at least 5%. In the present method, theoretical weak beam calculations indicated that a visible change in the CBED disc occurs in a foil of thickness 150 nm if thickness changes by 5nm. This suggests a potential accuracy of 3 %. However, the experimental uncertainty in the present method is estimated to be ,~ 6-7°/0. This is due to inaccuracies arising from experimental factors such as accelerating voltage, beam convergence, sample heating under the beam, and values of Fourier coefficients of the crystal potential. This 6-7% experimental uncertainty is expected provided that thickness is obtained by matching simulated multibeam discs to experimental observations. However, when Eqn (7) is used, experimental uncertainty may become --,10% or more due to the approximations inherent in deriving this formula. Acknowledgements--The financialsupport of CANDU owners Group,
Working Party 33, Work Package 3330 is greatly appreciated. The author is grateful to R. E. Mayville for experimental and technical assistance, and to C. D. Cann, R. A. Ploc, and S. S. Sheinin for discussion and helpful comments.
REFERENCES Botros, K. Z. and Cann, C. D., 1995. Thickness determination of Zr foils by convergent beam electron diffraction using the (11-20) reflectionat 300 keV. Atomic Energy Canada Ltd Report RC-541, in preparation. Cann, C. D., 1978. Convergentbeam thickness determination of thin foil zirconium specimens. Atomic Energy Canada Ltd Report AECL-5999. Cockayne, D. J. H., Ray, I. L. F. and Whelan, M. J., 1969. Investigations of dislocation strain fields using weak beams. Phil. Mag., 20, 1265. Cockayne, D. J. H., 1972. A theoretical analysis of the weak-beam method of electron microscopy.Z. Naturforsch., 27a, 452-460. Head, A. K., Humble, P., Clarebrough, L. M., Morton, A. J. and Forwood, C. T., 1973. Computed Electron Micrographs and Defect Identification. ElsevierScienceInc., New York. Hertel, B., Katerbau, K. H. and Bair, R., 1965. A possible source of error in the determination of the type of small lattice defects from electron micrographs. Phys. Stat. Sol. (A), 41, K1-K4. Hirsch, P. B., Howie, A., Nicholson R. B., Pashley D. W. and Wbelan M. J., 1977. Electron Microscopy of Thin Crystals. R. E. Kreeger Publishing Co., Marabar, Florida. Kelly, P. M., Jostsons, A., Blake, R. G. and Napier, Z., 1975. The determination of foil thickness by scanning transmission electron microscopy. Phys. Stat. Sol. (A), 31, 771-780.
Pergamon 0968-4328(95)00008-X
A Method for Determining Foil Thicknesses in TEM by Using Convergent Beam Electron Diffraction Under Weak Beam Conditions K. Z. B O T R O S
Fuel Channel Components Branch, Reactor Materials Division, Atomic Energy of Canada Ltd, Chalk River Laboratories, Station 55, Chalk River, Ontario, Canada KOJ 1JO (Received 14 November 1995)
Al~traet--In the present work, convergent beam electron diffraction was studied in zirconium (a material of intermediate atomic number) at 300 keV, under weak beam diffraction conditions. For a particular thickness, the details in an observed low order disc were matched to those calculated using the multibeam dynamical theory. This presents the possibility of determining foil thickness over a wide range, with an estimated experimental accuracy of ~ 7"/'0or less. In contrast to other convergent beam techniques, the present method, which uses weak beam conditions, can employ commonly-occurringlow order reflections to extract thicknesses. A simple equation based on the two beam approximation, is derived to determine foil thickness (to within ~10%) without resorting to detailed image matching. This equation can be used for a rough estimate of foil thickness while carrying out TEM observations. Copyright © 1996 Elsevier Science Ltd. Key words: foil thickness determination, convergent beam electron diffraction, weak beam conditions.
I. I N T R O D U C T I O N Foil thickness is an important parameter in m a n y analytical electron microscope (AEM) investigations, e.g. in assessing the density and nature of dislocations, in obtaining accurate micro-chemical analyses, and in studies of radiation-induced defects. Small changes in foil thickness can strongly affect defect contrast (see for example Hertel et al., 1977); and therefore accurate thickness determination is essential in correct defect identification (see, for example, H e a d et al., 1973). Kelly et al. (1975) have developed a method to determine foil thickness. This method employs convergent beam electron diffraction (CBED) when a low order reflection, g, satisfies its Bragg condition. The pattern in the g-disc consists of fringes which are normally symmetrical a b o u t the center of the disc, and foil thickness can be extracted f r o m the positions of such fringes. A criterion was outlined by Kelly et al. (1975) for the order of the reflection to be used. These authors warned of using some low order reflections, because systematic reflections can give rise to a complex C B E D pattern which m a k e thickness determination difficult. It is a c o m m o n practice to operate a modern A E M at its highest electron energy (200-400 keV) to take advantage of the improved resolution and greater penetration. At such electron energies, and especially in materials of intermediate atomic number, the effects of systematic reflections are expected to be quite strong. Consequently, problems can arise in extracting foil thickness, if the method proposed by Kelly et al. (1975) is used. A possible solution is to employ progressively higher order reflec-
tions (Cann, 1978). However, high order reflections lead to larger extinction distances which narrows the range of foil thicknesses that can be determined. It is possible to avoid the difficulties with strong effects of systematic reflections by choosing weak beam diffraction conditions (Cockayne et al., 1969), because diffraction is essentially kinematic under such conditions. The present work explored this possibility by studying the nature of the C B E D pattern in a low order disc obtained at 300 keV, under weak b e a m diffraction conditions. F r o m the pattern in such discs, extracting the foil thickness was explored in a material of intermediate atomic number, namely hcp zirconium. The nature of the C B E D pattern in the g-disc, resulting under weak b e a m conditions, is presented in section 2, and experimental examples are given in section 4.
H. N A T U R E OF T H E CBED P A T T E R N I N A G-DISC O B T A I N E D U N D E R W E A K B E A M DIFFRACTION CONDITIONS F o r a given incident b e a m direction, the amplitude of a diffracted beam ~bs (t), at the b o t t o m surface of a crystal of thickness t, can be written using the Bloch wave formulation (Hirsch et al., 1965) of the dynamical theory of electron diffraction as, N ~)g(t) : ~ e(0k)e(k) exp(27~i?(k)t) k=l 539
(1)
540
K.Z. Botros
where N in Eqn (1) is the number of systematic reflections considered, and it is also the number of Bloch waves which are excited. Assuming all Bitch waves to propagate along the zdirection, the flh Bitch wave will have the form b(J) = E
C(gj) exp(2qbi?(J)t)
(2)
g
In Eqn (2), cg03 is the b o l d {g}th element of the Bloch wave, and ~(/)= (kz0) - K z ) , where k~) is the z-component of the Bloch wave vector, Kz is the z-component of the inddent electron wave vector after correction due to the mean inner crystal potential. Under weak beam diffraction conditions, there are mainly (Cockayne, 1972) two Bloch waves (i and j ) contributing to the diffracted beam amplitude. Under such conditions, Bitch wave symmetry necessitates (Cockayne, 1972) that: C i)s'(i) O~g :
--°(J)e(J) = C "0 g
(3)
Substituting Eqn (3) in Eqn (1) gives fl~g(t) ~
cexp(2ni~(i)t)(1 - exp(2ni(? O) - ?(i))t))
(4)
The extinction distance under weak beam conditions effectively (Cockayne, 1972) given by ~g ~ (7 (]) --y(O) -1, and when this is substituted into equation (4), the intensity of the diffracted beam Ig, will be
is
Ig(t) ~ c2(1 - cos(2rct/~g))
(5)
Substituting for the extinction distance, ~g, its approximate value ~ US e under weak beam diffraction conditions (see Cockayne et al., 1969; Cockayne, 1972) gives
Ig( t) ~-. c2(1 -- cos(2nSgt) )
(6)
The condition for a minimum in the diffracted beam intensity is when (Set) is an integer, and for a maximum when (Set) has a value of n/2 (where n is an odd integer). Thus Eqn (6) predicts the pattern in the g-disc to consist of a number of parallel fringes with intensity alternating from bright to dark (maxima and minima). Equation (6) also predicts that the g-disc will exhibit high fringe contrast, because the maximum intensity is ~2e 2 while minimum intensity is almost zero. There will be a spread ASg in the values of the deviation parameter across a CBED disc from a minimum value Stain, to a maximum value of Smax- If two dark fringes occur at the edges of the disc, the first will occur when (tSmin) is a given integer, and the last is determined by (tSmax) being another integer. The number of dark fringes, m, in the g-disc will then relate Stain, Sm,x to the foil thickness t, through a simple relationship: t = m(A Sg) -1
(7)
In a general case dark fringes might not occur at the edges of the disc, i.e. at S~n or Sm~,. In such a situation,
a fraction should be added to the integral number of dark fringes counted in the disc. Thus, m, in Eqn (7) may become a real number to take into account a bright fringe which might appear at the edge of the disc. Equation (7) suggests that for a foil of thickness t, the g-disc will contain a number of fringes which only depend upon the difference (S,~x-Stain). If the width ASg of the CBED disc is measured and the number of fringes in it are appropriately counted, then Eqn (7) can be used to obtain the foil thickness.
m . MATERIALS AND M E T H O D S Two types of zirconium foils were employed in the present study. The first was prepared from material of ultra high purity, and the other from a Zr-2.5Nb pressure tube material used in C A N D U (CANadian Deuterium Uranium) reactors. All the CBED observations were carried out using a Philips CM30 electron microscope fitted with a twin objective lens, a double tilt holder, and operated at 300keV. Initially, a small second condenser aperture (usually 30mm) was used, and for a particular sample area chosen, tilting was carried out so that mainly one row of systematic reflections could be seen on the microscope screen. The sample was further tilted, so that the lowest order reflection in the systematic row had its deviation parameter, Sg, corresponding to weak beam diffraction (i.e. Sg>10-1nm -1 as defined by Cockayne, 1972). Caution was exercised to ensure that no low order nonsystematic reflections were close to their Bragg condition. The incident electron probe was then made convergent. The range of convergence angles 20is was 5 x l 0 - 3 - 1 2 x l 0 - 3 r a d . This was achieved by using different sizes of the second condenser aperture. The resulting CBED pattern was recorded on a photographic plate using an appropriately large camera length for clarity. The angular spread in a recorded CBED pattern depends on the beam convergence angle, 2~s, which gives rise to a spread ASg in the deviation parameter Sg, spanning values from a maximum Sm~x, to a minimum Stain. The center of the g-disc will be at a deviation Sg=(Smm+ASg/2). The spread ASs, depends on the Bragg angle of the reflection employed. A range of 'widths' for the g-disc were explored (see Table 1 for typical values for some commonly occurring reflections in Zr). The central value of Sg (corresponding to the middle of the disc) was determined from the position of the Kikuchi lines. Under weak beam diffraction conditions these lines Table 1. Typical range of 'width' in the convergent beam disc explored for different reflections in Zr (plane spacing are given). Disc width is expressed in fractions of 08, the corresponding Bragg angle Reflection
d (nm)
Range of disc width explored
0002 11-20
0.258 0.162
~1.4 00oo2to ~1.8 0ooo2 ~1.2 011-2o to ,-d.6 011-2o
Foil Thickness in TEM
were located with higher accuracy than is possible when using strong beam diffraction conditions, as required by Kelly et al., 1975). The spread in 6Sg in a recorded g-disc was determined from its diameter and its distance to the 0disc (after suitable calibration of angular spread). Accuracy in determining Sg, and 6Sg were estimated to be ~ 2 % , compared with ~ 5 % reported by Cann (1978), under strong beam diffraction conditions. The values of 6Sg and Sg were used to calculate the intensity of a particular diffracted beam against angular deviation. Intensity was converted to 256 gray levels, thus generating a theoretical CBED pattern for a more meaningful comparison to experimental observations. The CBED-patterns calculated by assuming different thicknesses showed sensitivity to small variations in foil thickness. An example of this is shown in Fig. la,b where visible changes in the CBED pattern (e.g. in the number of dark fringes) occur due to a small foil thickness variation. Thus to determine a foil thickness in the present work, thickness was varied in the calculation until good agreement was obtained between theory and experimental CBED patterns. The present calculations were based on the scattering matrix approach of the dynamical theory of electron diffraction (see Hirsch et al., 1965), which assumes the column approximation. The effects of absorption were taken into account by assuming the ratio of the imaginary to real part of the lattice potential to be 0.1. The value of this ratio was found to be not critical, as calculations showed that ratios between 0.06 and 0.12 gave no discernible effect on the simulated CBED patterns.
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Fig. 2. The grain structure in a longitudinal section (i.e. foil normal is along the length of the tube) of Zr-2.5 Nb pressure tube. Most of the HCP grains in such material (such as those marked 'A') give diffraction patterns which contain the (0002) reflection.
in a g-disc when weak beam diffraction conditions are chosen. In this section, examples of observations in Zr2.5Nb pressure tube samples are presented. The agreement between observations and results of theoretical calculations are also presented. HCP ct-Zr grains in pressure tube material exhibit strong texture. For example, in a longitudinal section
(a)
IV. RESULTS OF OBSERVATIONS AND THICKNESS DETERMINATION. Arguments in Section 2 point to the possibility that foil thickness can be determined from the CBED pattern
(a)
(b)
.
(b)
Fig. 1. Simulated multibeam CBED patterns in Zr assuming the (11-20) reflection satisfying weak beam diffraction conditions at 300 keV. (Sm~-Sn~n) was assumed 1.6ql 1-2o. The foil thickness assumed in (a) was 95rim, and for Co) 99nm. Such a small change in thickness can be seen to give rise to visible changes in the CBED pattern. The resulting dark fringes are numbered.
~'~
:: ~
~
~.
O000-disc
O002-disc
Fig. 3. (a) Bright field and dark field CBED discs observed in a foil region from a longitudinal section of Zr-2.5Nb pressure tube. Observations were carried out at 300 keV and used the (0002) reflection under weak beam diffraction. Each disc had an angular width ~ 1.50o0o2. The deviation parameter at the center of the 0002-disc was S00o2~1.03 x l0 - l nm -1. Co) Simulated multibeam CBED discs calculated under the same conditions leading to the convergent beam pattern in (a). The foil t h i c k n e s s assumed was 240 nm. Note the weak contrast in the bright field disc.
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K . Z . Botros
(with a normal along the length of the pressure tube), the majority of =-grains (some are marked as 'A' in Fig. 2) have normals near the (2-1-10) and (01-10) directions. The diffraction patterns of these foils frequently contain the low order (0002) reflection. It was then important to explore the use of this reflection to obtain a CBED pattern under weak beam conditions. An example is shown in Fig. 3a. In this figure bright field and dark field discs are shown. While such pattern was being observed on the microscope's screen, the bright field 0000-disc exhibited poor contrast and was of much higher intensity than the dark field 0002-disc which exhibited high contrast. The exposure time required to record the details of the dark field disc on the photographic plate, gave an overexposed bright field disc (see Fig. 3a). In Fig. 3b, results of calculations are presented. These calculations assumed the experimental diffraction conditions in obtaining Fig. 3a, and took into account the effects of 15 systematic reflections. The bright field disc in Fig. 3b shows the poor contrast that was experimentally observed, and because of this, no calculated bright field discs will be subsequently presented. On the other hand, many observed 0002discs from different foil regions were compared to theory. In each case good matching could be obtained between calculated and experimental 0002-discs, for an
appropriate foil thickness. An example of this can be seen in Fig. 3a,b. Weak beam observations were also carried out for other reflections in Zr. In Fig. 4a, an experimental example is shown for the (11-20) reflection. Fig. 4b shows the result of the corresponding multibeam calculation. Again, excellent agreement between observation and results of theoretical multibeam calculations can be seen by comparing Fig. 4a to Fig. 4b. This type of agreement was obtained between theory and experiment for a variety of observations covering a wide range of foil thicknesses (~ 35-~300 rim). The two beam theory could also be used to match experimental observations. The predicted thickness was, however, slightly different from that obtained from multibeam calculations. For example, good agreement between simulated two beam calculations and the experimental (0002) disc shown in Fig. 3a was obtained when foil thickness was assumed to be ~250 rim. This is only ~ 4 % different from the thickness predicted by multibeam calculations. Smaller differences were seen in other cases involving higher order reflections. The 4% difference found for the (0002) case, was nearly equal to that between the two beam and the multibeam extinction distances calculated under weak beam conditions. Finding that good agreement can exist between ob-
(a)
(b)
1120-disc Fig. 4. (a) C B E D diffraction pattern in a Z r - 2 . 5 N b foil obtained at 300keV using the (11-20) reflection under weak b e a m diffraction conditions. The angular width o f each o f the observed bright, dark field discs was ~ 1.5011_20. The deviation parameter at center o f the 11-20-disc was S]1_2o~2.6 x 10 - I n m - ] , a n d ( S m ~ - S m j , ) ~ 0 . 5 6 x 10 -~ n m - I . Co) Simulated multibeam 11-20disc, calculated under the same conditions leading to the convergent b e a m pattern in (a). The foil thickness a s s u m e d was 226 urn.
Foil Thickness in TEM servations and two beam calculations illustrates that under weak beam conditions, contrast in CBED pattern arises mainly from the interaction of two Bloch waves, as indicated in Section 2. Equation (7) in Section 2 offered the possibility of thickness determination without resorting to image matching. This is illustrated by an example for the (11-20) case in Fig. 4a. The l l-20-disc has a width (Smax--Smin) ~ 0 . 5 6 x 1 0 - 1 n m -1, and can be seen to contain 12.5 dark fringes. The thickness calculated from Eqn (7) is ~ 2 2 3 n m , as compared with ~ 2 2 6 n m obtained by a detailed image matching procedure. Such agreement indicates the possibility of using Eqn (7) for foil thickness determination.
V. D I S C U S S I O N Foils in the thickness range 30°300 nm are useful in studies of lattice defects in a material like Zr and Z r 2.5Nb at intermediate electron energies. The present work shows that specimen thickness can be determined when the experimental CBED pattern obtained under weak beam conditions is in good agreement with details o f a simulated disc. This method, discussed in Sections 2 to 5, offers the following possibilities: (a) Permits defect observations and thickness determinations over a wide range while using the microscope at intermediate incident electron energies (200-400 keV); (b) Easy application in foils of highly textured materials. The method can employ commonly occurring reflections, eliminating the need for complicated tilting procedures searching for suitable reflections; (c) Estimating thickness during A E M observations without resorting to detailed image matching procedures, can be done by counting the number of fringes in the CBED disc and using Eqn (7). There are limitations in using the method of Kelly et al. (1975) at intermediate electron energies for materials o f intermediate atomic number. At 200keV, Cann (1978) showed that the (11-20) is the lowest order reflection to employ in Zr to determine foil thickness. At 300 keV, Botros and Cann (1995) showed that if the (1120) reflection is chosen, the correct thickness determination requires the multibeam extinction distance, and that such determination is limited to a narrow range of ~1500250nm. In contrast to this, the present method offers the possibility of determination over a wider range, 30-300 nm. An important condition in employing the present method, is the choice of weak beam orientations such that no low order non-systematic reflection is close to its Bragg condition. This may be difficult to achieve in foil regions which are excessively thin and bent. In addition,
543
when accelerating voltages in excess of ,-~500 keV are used, difficulty in employing the present method may arise, at least in utilizing low order reflections like (0002) in Zr. This may be due to problems in realizing good weak beam conditions and in avoiding effects of nonsystematic reflections. Finally, it is important to comment on the experimental uncertainty of thickness determination expected from different convergent beam methods. Kelly et al. (1975) gave an optimistic ,-~2% in copper, while Cann (1978) estimated experimental uncertainty in zirconium to be at least 5%. In the present method, theoretical weak beam calculations indicated that a visible change in the CBED disc occurs in a foil of thickness 150 nm if thickness changes by 5nm. This suggests a potential accuracy of 3 %. However, the experimental uncertainty in the present method is estimated to be ,~ 6-7°/0. This is due to inaccuracies arising from experimental factors such as accelerating voltage, beam convergence, sample heating under the beam, and values of Fourier coefficients of the crystal potential. This 6-7% experimental uncertainty is expected provided that thickness is obtained by matching simulated multibeam discs to experimental observations. However, when Eqn (7) is used, experimental uncertainty may become --,10% or more due to the approximations inherent in deriving this formula. Acknowledgements--The financialsupport of CANDU owners Group,
Working Party 33, Work Package 3330 is greatly appreciated. The author is grateful to R. E. Mayville for experimental and technical assistance, and to C. D. Cann, R. A. Ploc, and S. S. Sheinin for discussion and helpful comments.
REFERENCES Botros, K. Z. and Cann, C. D., 1995. Thickness determination of Zr foils by convergent beam electron diffraction using the (11-20) reflectionat 300 keV. Atomic Energy Canada Ltd Report RC-541, in preparation. Cann, C. D., 1978. Convergentbeam thickness determination of thin foil zirconium specimens. Atomic Energy Canada Ltd Report AECL-5999. Cockayne, D. J. H., Ray, I. L. F. and Whelan, M. J., 1969. Investigations of dislocation strain fields using weak beams. Phil. Mag., 20, 1265. Cockayne, D. J. H., 1972. A theoretical analysis of the weak-beam method of electron microscopy.Z. Naturforsch., 27a, 452-460. Head, A. K., Humble, P., Clarebrough, L. M., Morton, A. J. and Forwood, C. T., 1973. Computed Electron Micrographs and Defect Identification. ElsevierScienceInc., New York. Hertel, B., Katerbau, K. H. and Bair, R., 1965. A possible source of error in the determination of the type of small lattice defects from electron micrographs. Phys. Stat. Sol. (A), 41, K1-K4. Hirsch, P. B., Howie, A., Nicholson R. B., Pashley D. W. and Wbelan M. J., 1977. Electron Microscopy of Thin Crystals. R. E. Kreeger Publishing Co., Marabar, Florida. Kelly, P. M., Jostsons, A., Blake, R. G. and Napier, Z., 1975. The determination of foil thickness by scanning transmission electron microscopy. Phys. Stat. Sol. (A), 31, 771-780.