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Application of microdiffraction to crystal structure identification
ultramicroscopy ELSEVIER
Ultramicroscopy53 (1994) 305-317
Application of microdiffraction to crystal structure identification A. Redja'imia a, j.p. Morniroli b a Laboratoire de Science et G~nie des Surfaces, URA CNRS 1402, Ecole des Mines, Parc de Saurupt, F-54042 Nancy Cedex, France Laboratoire de M~tallurgie Physique, URA CNRS 234, Universit~ de Lille I, Bdtiment C6, F-59655 Villeneuue d'Ascq Cedex, France
(Received 25 January 1993; in final form 28 December 1993)
Abstract
In a previous paper by J.P. Morniroli and J.W. Steeds [Ultramicroscopy 45 (1992) 219] it was shown that microdiffraction patterns obtained with a nearly parallel electron beam and a small spot size can be used to identify and to characterize crystal structures. A systematic method was proposed to deduce, via the Buerger extinction symbol, the space group or a set of consistent space groups. In order to illustrate how this method can be applied experimentally, three examples concerning the ~, g and R intermetallic phases are detailed. Experimental solutions are given to overcome difficulties connected with the frequent absence of high-order Laue zone reflections on high-symmetry zone axis patterns. The possibility to identify the Buerger extinction symbol from a unique zone axis pattern is also explored.
1. Introduction
In a previous paper by J.P. Morniroli and J.W. Steeds [1] it was shown that microdiffraction patterns obtained with a focused and nearly parallel electron beam allows one to obtain, in a systematic way, the crystal system, the Bravais lattice and to reveal the presence of glide planes. These crystallographic features are easily deduced from observations of three pieces of information present on the microdiffraction patterns, namely: • The " n e t " symmetry, that is the 2D plane symmetry which is concerned with the position of the reflections on the microdiffraction pattern. For each diffraction pattern, the " n e t " symmetry of the zero-order Laue zone ( Z O L Z ) and the " n e t " symmetry of the whole pattern (WP) can be inferred. Both symmetries are in connection with the crystal system.
• The shift between the Z O L Z and the first-order Laue zone (FOLZ) reflection nets. It is in connection with the Bravais lattice [2,3]. • The periodicity difference between the Z O L Z and the F O L Z reflection nets. It is in connection with the presence of glide planes [4-6]. A systematic method is proposed which involves three steps. The deduction of the crystal system constitutes the first step. It is made by searching the experimental microdiffraction pattern which exhibits the highest " n e t " symmetries. The crystal system is then deduced by means of Table 7 in Ref. [1] and the corresponding zone axis by means of Table 2 in Ref. [1]. In a second step, a simultaneous determination of the Bravais lattice and glide planes is made by comparing microdiffraction patterns with theoretical ones. For each crystal system, one,
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A. Redja~mia, J.P. Mornirofi / Ultramicroscopy 53 (1994) 305-317
two or at most three different experimental patterns corresponding to high crystal symmetry elements are required for this analysis, as indicated in Table 5 in Ref. [1]. All these patterns, except for those connected with the monoclinic system, contain at least two orthogonal " n e t " mirrors m, and m 2. In order to facilitate the comparison with the theoretical patterns two features are taken into account: • The smallest rectangles or squares, centered or not, and having their sides parallel to the " n e t " mirrors m I and m 2 are drawn in the Z O L Z and in the F O L Z . They give information about the Z O L Z / F O L Z periodicity difference connected with the presence of glide planes. • The presence or the absence of F O L Z reflections situated on these mirrors m~ and m 2. They give information about the Z O L Z / F O L Z shift connected with the Bravais lattice. • Eventually screw axes can also be identified by specific methods described in Ref. [1]. Comparison with the theoretical patterns leads to the extinction symbol introduced by Buerger [7], which by means of Table 3.2 of the International Tables for Crystallography [8] allows one to deduce a few possibilities of space groups. These experiments are usually easy to perform since they do not require a very accurate orientation of the specimen with respect to the electron beam. They can be carried out successfully with most specimens. The third and last step consists of removing ambiguity between the possible space groups. Depending on the specimen, two ways are possible. If the specimen gives good quality C B E D (convergent b e a m electron diffraction) patterns displaying 3D or at least 2D information, then, these patterns can be used to determine the point groups by means of the methods given by Buxton et al. [9] or by Tanaka et al. [10]. It is pointed out, that the results previously obtained by microdiffraction can be used, in association with Tables 2 - 4 in Ref. [9] or Fig. 3 in Ref. [10], to plan the simplest C B E D experiments to perform. Very often, C B E D patterns only displaying 2D information are sufficient to realize this analysis and this constitutes a very important experimental simplification.
If good C B E D patterns are not available, then an alternative solution consists of examining the "ideal" symmetries of the microdiffraction patterns. The "ideal" symmetry is the 2D plane symmetry which takes into account both the position and the intensity of the reflections on a microdiffraction pattern. As indicated in Table 8 in Ref. [1], this "ideal" symmetry is connected with the point groups and, from observation of specific zone axes, it can be used to identify the point group. Nevertheless, the corresponding experiments are sometimes difficult to perform since they require a perfect orientation of the crystal with respect to the electron beam. In this paper it is intended to show that this approach can be used as a routine technique to identify and characterize phases. To that aim, the examples of the or, X and R intermetallic phases are given. All the notations and the conventions given in Ref. [1] will be used in this paper. Particularly, the "ideal" symmetries and mirrors will be underlined in order to make the distinction with the " n e t " ones and all features related with the Z O L Z will be written between parentheses. For patterns with 2mm " n e t " symmetry containing two perpendicular mirrors, the mirror parallel to the larger side of the rectangle drawn in the F O L Z is named m I and the other m 2. For patterns with 4mm " n e t " symmetry containing two sets of perpendicular mirrors, the two perpendicular mirrors parallel to the smallest square drawn in the F O L Z are named ml, m2 and the other set m' 1, m 2. For patterns with 6mm " n e t " symmetry, the set of three mirrors parallel to the shortest reflections present in the Z O L Z is named m,, m2, m 3 and the second set, rotated at 30 ° with ' m ' 3respect to the first one, is named m'~, m2, Trigonal crystals are described by means of a triple obverse hexagonal cell. !
2. Experimental methods The tr, X and R intermetallic phases observed in this study are present in a duplex austeniticferritic stainless steel as small particles with an average size of about 0.1 /xm [11]. Thin foils for
307
A. Redjafmia, J.P. Mornirofi / Ultramicroscopy 53 (1994) 305-317
Incident beam [uvw] Zone Axis Specimen
29~. FOLZ diffracted beams rransm~ed
O*
Tiltajn lie
22.
sphere
~La~er +1 er 0 Layer-1 a 2%,.
Incident beam ~i
~
/
FOLZ
~iffl"actedbeams
Layer -1
b
L
c
Fig. 1. Ewald sphere constructions: (a) The incident beam is parallel to a zone axis giving a symmetrical zone axis pattern. (b) The incident beam is slightly tilted from the zone axis in order to improve the visibility of the FOLZ reflections. (c) The incident beam is tilted with a 20syrn angle from the zone axis so that the first negative layer of the reciprocal lattice becomes tangential to the sphere. A "FOLZ" diffraction pattern is obtained.
A. Redjai'mia, J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317
308
S'¢mm,etrles {N = "Neet" : I = "Ideal')
Sigma
a = I0011
(4,~nt) WP
4ram
b = II(YOl
J ~
e=lllOI N ] (2rnnl) ~ ) 2ran___.! 2ram 2mn.....~l
(2~,n)
I
i 2111111
Extinction Symbol
P-..
P. n
P_-
"~t°
-•Ol ij.....m2
a
[oou ZAP
m'oo,
~
[I 10] Z A P
002 :
. ~il,.m2: t)
{ lOOl Z A P
m: 110
A. Redjai'mia, J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317
electron microscopy are prepared by twin jet electrothinning in a solution of 5% perchloric acid in 95% II-butoxyethanol at 40 V potential. They are observed with the Philips CM12 and CM30 electron microscopes. The small convergence angle is obtained by using a 10 /zm C 2 condensor aperture and the spot size is in the range 10-100 nm. The zone axis patterns are accurately oriented with respect to the electron b e a m by using the dark field deflectors. Frequently, high-symmetry zone axis patterns exhibit large F O L Z rings. As a result, the F O L Z reflections are very weak due to their large 0sy m diffraction angle as shown on the Ewald sphere construction of Fig. la or absent due to the limited acceptance angle of the microscope. Therefore, the " n e t " and "ideal" whole pattern symmetries, as well as the Z O L Z / F O L Z shifts and periodicity differences are not available on these symmetrical patterns. To solve this problem, the operating voltage of the microscope can be reduced so as to decrease the F O L Z radius or the specimen can be cooled to liquid-nitrogen t e m p e r a t u r e to enhance the intensity of the F O L Z reflections. A better solution consists in tilting the specimen slightly around the zone axis until F O L Z areas appear on the pattern (Fig. lb). Due to lower 0 diffraction angles, the F O L Z reflections obtained in that way are usually stronger and more visible. The tilt angle can be chosen so that it equals 20sym. Then, the first negative layer plane of the reciprocal lattice becomes tangential to the Ewald sphere as indicated in Fig. lc and a " F O L Z " microdiffraction pattern is obtained. This experimental solution, which was previously used by Steeds et al. [12,13] to analyze the fine structure of H O L Z reflections, constitutes an elegant
309
Fig. 3. [001] " F O L Z " microdiffraction pattern of o- phase obtained by tilting the incident beam around [001] as indicated in Fig. lc.
method to characterize the Z O L Z / F O L Z and periodicity differences.
shifts
3. Experimental results 3.1. Example I: o" phase The major Z A P s investigated for this phase show that the highest " n e t " symmetry encountered is (4mm), 4mm (Fig. 2a). This pattern was obtained at 120 kV from a specimen cooled at liquid-nitrogen temperature. The F O L Z reflections are clearly observed. At 200 or 300 kV these F O L Z reflections are not visible on a symmetrical pattern but they can be observed by tilting. A " F O L Z " microdiffraction pattern obtained by the method described in Section 2, is given in Fig. 3.
Fig. 2. Microdiffraction patterns for the tetragonal o- phase: (a) [001] zone axis microdiffraction pattern showing (4mm), 4 m m " n e t " and (4mm), 4 m m "ideal" symmetries. The absence of Z O L Z / F O L Z periodicity difference indicates that there is no glide plane perpendicular to the [001] direction and that the partial extinction symbol is P - . . . (b) [100] zone axis microdiffraction pattern showing (2mm), 2mm " n e t " and (2mm), 2mm "ideal" symmetries. T h e periodicity difference between the Z O L Z and the F O L Z reflection nets reveals the presence of a diagonal n glide plane perpendicular to [100]. The deduced partial extinction symbol is P • n •. (c) [110] zone axis microdiffraction pattern showing (2mm), 2mm " n e t " and (2mm), 2mm "ideal" symmetries.The absence of Z O L Z / F O L Z periodicity difference indicates that there is no glide plane perpendicular to the [110] direction and that the partial extinction symbol is P . . - .
A. Redja'[rnia, J.P. Morniroli / Ultrarnicroscopy 53 (1994) 305-317
310
Since no pattern with (6mm) Z O L Z " n e t " symmetry was observed, the crystal system is, according to Tables 7 and 2 in Ref. [1], tetragonal and the zone axis with (4mm), 4mm " n e t " symmetry is [001]. For this crystal system, the ZAPs needed to identify the Bravais lattice and the glide planes are [100], <100) and <110) (Table 5 in Ref. [1]). To compare these experimental patterns with the theoretical ones of Fig. 9c in Ref. [1] the following operations are performed: • The " n e t " mirrors are identified and the F O L Z reflections present on them are distinguished by means of open circles.
• The smallest squares or rectangles (centered or not) whose sides are parallel to these mirrors are drawn both in the F O L Z and in the Z O L Z . For the F O L Z , this operation is very easy to perform on the " F O L Z " microdiffraction pattern of Fig. 3. The [001] pattern (Fig. 2a) contains two sets of perpendicular " n e t " mirrors ml, m 2 and m'1, m'2. F O L Z reflections are situated along each of these mirrors. The smallest squares whose sides are parallel to m], m2 are identical in the F O L Z and in the Z O L Z . The (100) pattern (Fig. 2b) exhibits F O L Z reflections along the two " n e t " mirrors m 1 and
/~i . . . . . . . . . .
":::" ,;:::::~:::::, ~::: ,. : ::": " : :.::::": ; ' " . . . Jmj"::::" . . . " : : : : ":: " : : "~'\ ::: :::: .::::~:. :~: ~:\ :: : : : :::::~:::: ::: :" |oe
•
• ee*
i : : " :::"
• • * * * *i*e~:e**eo
::::":::::
m~ • -**a
,e,l *e~
m2.
"::: "::
\'i!.'~!!:. ":::':::" .:~W.ii ~]
[oo~] zAP
P-n- or P-b-
a
b
Fig. 4. I d e n t i f i c a t i o n of t h e f o r b i d d e n reflections by tilting c a r r e t u l l y t h e s p e c i m e n a l o n g t h e m i r r o r s m I a n d m 2 so as to o b s e r v e
parts of the HOLZs. The TO01 and ]~ 0 2 forbidden reflections are identified in the FOLZ and in the SOLZ along the mirror m 2.
A. Redja?mia, J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317 Table 1 Space groups and point groups for extinction symbol P-nExtinction symbol P-n-
Space groups
Point groups
P42nm P4n2 42 21 2 P. . . m n m
4mm 4m2 4 mm m
.
m 2. The centered rectangle drawn in the ZOLZ has double size as compared with the noncentered rectangle drawn in the FOLZ.
WP Symmetry ZAP I
311
The (110) pattern (Fig. 2c) only displays FOLZ reflections along the m 2 "net" mirror and the same rectangles, with sides parallel to m I and m2, are present in the ZOLZ and in the FOLZ. Comparisons of these three experimental ZAPs (Figs. 2a-2c) with Fig. 9c in Ref. [1] give the following individual extinction symbols: P--for [001], P- nfor (100), P'.for (110). The addition of these three results gives the partial extinction symbol P-n- which according to
2mm [1101 ]
Fig. 5. [110] C B E D pattern displaying 2D information inside the transmitted and diffracted discs and exhibiting a 2mm WP symmetry.
312
A. Redja't'mia, ZP. Morniroli/ Ultramicroscopy 53 (1994) 305-317
T a b l e 3.2 in Ref. [8] is in a g r e e m e n t with 3 space g r o u p s b e l o n g i n g to 3 p o i n t g r o u p s as i n d i c a t e d in T a b l e 1. It is also possible to o b t a i n the P-n- extinction symbol directly on t h e [001] Z A P . T h e t h e o r e t i c a l p a t t e r n which c o r r e s p o n d s to the extinction symbol P-n- (or P-b-) given in Fig. 4a displays forbidd e n reflections in t h e Z O L Z which can be identified by m e a n s of G j 0 n n e s a n d M o o d i e lines [14]. It also c o n t a i n s Okl a n d hOl f o r b i d d e n reflections l o c a t e d in t h e F O L Z a n d in the S O L Z (secondo r d e r L a u e z o n e ) along the " n e t " m i r r o r s m I a n d
m2. T h e s e f o r b i d d e n reflections c a n b e o b s e r v e d on the p e r f e c t l y o r i e n t e d [001] zone axis p a t t e r n . A m u c h b e t t e r solution consists of tilting the s p e c i m e n along the m i r r o r s by m e a n s o f the d a r k - f i e l d b e a m d e f l e c t o r s as shown in Figs. 4b a n d 4c. T h e F O L Z reflections a r e t h e n m o r e visible a n d t h e f o r b i d d e n r e f l e c t i o n s i d e n t i f i e d m o r e surely. T h e distinction b e t w e e n the two extinction symbols P-n- a n d P-b- can b e m a d e by indexing the Okl a n d hOl f o r b i d d e n reflections. In the first case, t h e f o r b i d d e n reflections a r e of t h e type
Symmemcs (N = "Net" ; I = "ldeaI") m'3 Khi ZOLZ WP Extinction Symbol
a = 10011 N [ I
b= Illll N
(4mm)[ (4_..~I1) (6ram) 4ram I 2ram
3m
c = 10111 N (2rean) 2ram
1-.. o r F-..
I..•
m2
,,.,
10 OmZAP
0 iii
;i ii ii~
2OO
a
I0011ZAP
\P
Fig. 6. Microdiffraction patterns of the cubic X phase: (a) [001] zone axis microdiffraction pattern showing (4mm), 4mm "net" and (4mm), 2mm "ideal" symmetries. The absence of ZOLZ/FOLZ periodicity difference indicates that there is no glide plane perpendicular to the [001] direction and that the partial extinction symbol is I-.. or F-... (b) [Ell] zone axis microdiffraction pattern showing (6mm), 3m "net" symmetries. (c) [011] zone axis microdiffraction pattern showing (2mm), 2mm "net" symmetries. The absence of ZOLZ/FOLZ periodicity difference indicates that there is no glide plane perpendicular to the [011] direction and that the partial extinction symbol is I . . - .
A. Redja'imia, J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317
(k + l) 4: 2n or (h + l) 4~ 2n while in the second case they are of the type k 4: 2n or h 4: 2n. The forbidden reflections identified in Fig. 4b have 10 0 1 and 19 0 2 indices. In addition, no forbidden reflections are observed along the mirrors m'1 and m'2 (Fig. 4c). These results are in perfect agreement with the P-n- extinction symbol. The distinction between the three possible space groups requires the identification of the point group. This identification can be inferred from observation of the "ideal" symmetries, that is the symmetries which take into account both the position and the intensity of the reflections on the microdiffraction pattern. The "ideal" symmetries for the [001] Z A P are (4mm), 4mm (Fig. 2a). According to Table 8 in Ref. [1], the possible point groups are then 4mm or 4 / m m m . To remove the ambiguity between these two point groups, observation of the "ideal" symmetry of the (100) or (110) ZAPs is required. The two experimental [100] (Fig. 2b) and [110] (Fig. 2c) ZAPs exhibit (2mm), 2mm "ideal" symmetries. Therefore it is concluded that the point and space groups for the or phase are respectively 4 / m m m and P 4 2 / m 2 1 / n 2 / m . This phase provides an example where microdiffraction can also be used in connection with C B E D to identify the point group. For this phase it was possible to obtain (110) C B E D patterns which display 2D information inside the transmitted and diffracted discs (Fig. 5) and exhibit a 2mm whole pattern symmetry. Since only 2D information is present, the projection diffraction groups must be taken into account. From Tables 3 and 4 in Ref. [9] we can deduce the projection diffraction groups and then the corresponding whole pattern symmetry for the 3 possible point groups as given in the Table 2. The 2mm WP symmetry observed on the experimental pattern of Fig. 5 is in agreement with the 4 / m m m point group. Therefore the space group for the ~ phase is P 4 z / m 2 t / n 2 / m . It is pointed out that a C B E D pattern which only exhibits 2D features inside the discs allows one, in connection with microdiffraction patterns, to identify uniquely the point and space group of this phase. Since C B E D with 2D features are easier to obtain than C B E D patterns with 3D
313
Table 2 Projection diffraction group and whole pattern symmetry Point group Projection WP symmetry Diffraction group 4mm m1R m 7,2m ml R m 4 -- mm 2mm1R 2mm m
information, this constitutes an important experimental simplification.
3.2. Example II." X phase The highest " n e t " symmetries observed for this phase are (4mm), 4mm (Fig. 6a) and (6mm), 3m (Fig. 6b) which, according to Table 7 in Ref. [1], correspond to a cubic system. One notices that the identification of the cubic system does not necessarily require the observation of the 4mm and 3m whole pattern " n e t " symmetries. This system can be identified, in a simplified way, only from the (4mm) and (6mm) Z O L Z " n e t " symmetries. The zone axes with (4mm), 4mm and (6mm), 3m symmetries respectively correspond to the (001) and (111) Z A P s as indicated in Table 2 in Ref. [1]. The specific Z A P s to observe for Bravais lattice and glide plane identifications are (001) (Fig. 2a) and (110) (Fig. 2c) (Table 5 in Ref. [1]). On the (001) ZAP, two sets of perpendicular " n e t " mirrors ml, m 2 and m'~, m'2 are identified. The smallest squares drawn in the Z O L Z and in the F O L Z have their sides parallel to the m~, m 2 mirrors and are identical. F O L Z reflections are present on the m'~, m'2 mirrors but absent on ml, m 2 mirrors. In agreement with Fig. 9d in Ref. [1], the corresponding individual partial extinction symbols is either I -.o or F - - . . Therefore, the distinction between the I and F Bravais lattices is not possible from observation of this unique (001) ZAP. The (110) Z A P (Fig. 6c) exhibits (2mm), 2mm " n e t " symmetries. No F O L Z reflections are present on either of the two perpendicular " n e t " m 1 and m 2 mirrors and the two rectangles with sides parallel to the mirrors drawn in the Z O L Z and in
314
A. Redja'fmia, J.P. Morn iroli/Ultramicroscopy 53 (1994) 305-317
the F O L Z are identical. Comparison with Fig. 9d in Ref. [1] leads to the individual partial extinction symbol I • .-. Contrary to the (001) ZAP, the body centered Bravais lattice is uniquely identified from this (110) ZAP. Addition of the two individual partial extinction symbols leads to I--- which is in agreement with the 7 space groups associated with the 5 point groups (Table 3.2 in Ref. [8]) given in Table 3. To make the distinction between these seven
possible space groups, the point group must be identified. This can be realized from observation of the (001) Z A P "ideal" symmetry as indicated in Table 8 in Ref. [1]. This (001) pattern (Fig. 6a) exhibits a (4mm) "ideal" Z O L Z symmetry. However, careful examination of the intensity of certain couples of reflections (see particularly the couples of reflections arrowed on Fig. 6a) reveals that the WP "ideal" symmetry is only 2mm since the two " n e t " mirrors m t and m 2 are not "ideal" mirrors if the intensity of the reflections is taken
OlT2~ OTll
13 [iltOlZAP
Synlnlctllex (N = "Net" : I = "ldeal"~
R phase
IT 2TOl
m2
........ : 1210 a
ZOLZ WP Extinction Symbol
a = 100011 (6n~mt
3m
I (6) ~
b =121101 N t21 2 RIob~ J
m]
[0001 ] Z A P
Fig. 7. Microdiffraction patterns of the trigona] R phase: (a) [0001] zone axis microdiffraction pattern showing (6ram) and 3m " n e t " and (6), _3 "ideal" symmetries. (b) [2110] zone axis microdiffraction pattern, showing (2) and 2 " n e t " symmetries. It reveals no periodicity difference between thc Z O L Z and the FOLZ. The deduced extinction symbol is R(obv)--.
A. Redjai'mia,J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317 Table 3 Space groups and point groups for extinction symbol I--Extinction Space groups Point groups 123 23 1213
I---
Im3
m3
I432 14132
432
I43m
43m
4_2 I--3-m m
4_2 --3-m m
5 in Ref. [1]). Analysis of the [ff.ll0] ZAP, which exhibits (2), 2 " n e t " symmetries (Fig. 7b), reveals no periodicity difference between the Z O L Z and
::::,:?:': :':':;*:!:: "" A:;:".. ' " " ::;:X • 0 • •
• •
"410 • O 0 0 O 0 0 0 l t / • 0"41 • O 0 O,O O 0 • (~"O • O
° o' e. •
oO00 • •
• OOII • • e
"i, • o!o • ; "......
......
• •
O0000
0 0 O
• 0
into account. According to Table 8 in Ref. [1], the point group corresponding to (4mm), 2mm "ideal" symmetries is 43m. It is concluded that the space group of this investigated X phase is I43m.
.."0
O O •
o o
.,"
• '
• • • co°ego
OOq)0 • •
.......ii!i!1
• O0000 • • •
O •
oo
O •
•
0
..
,."'el 0 0
\'.'..
0 O O O O • q)"..~
O O B 0 O0
.
•
:
-.. ,!._._._.'
0 @ • .," • 410,""
~.-~
¢-. .
01111
eo
/
/
!
• • •
'~..
o-~..o e
,_ _
~
\
\
/ ".'_.'..~G\ ."
.
".'A
~Oe
.,.v_.-_._ ".',~,X
--
i
•
I110
. - "-".v.\ "'o',
e e ~ o e e e o ' e o e e t
I ";'~.'.'." ', • e
~
'~
...':
O"O..
,%'7
, 0 • 0 " • • • 0 , - , - , ~
/::" ',, . . . .f. .. 0".-" 't,o,,'n,,,,¥
/
•
41 41
O0 • . "~ 0 •
m~' " Z ~
,~-.,.
o •
• •
00000~0090
/
O •
".
• 0
.
•
410
0 • O"...
• •.
0 O0 . 0 O
O
• oo
"\
,,,,: - - , 0 0 O0
O0
' "'"' O
~
."
3.3. Example III: R phase
Table 4 Space groups and point groups for extinction symbol R(obv)-Extinction symbol Space groups Point groups R3 3 R3 R(obv)-R32 32 R3m 3m R3m 3m
°..~i~°
ql0 • • • OoO004 ,, .-" oOoeoOeOoOo • • ' i • • •
0 • • •
.. j..:::::...............
O •
For the R phase, the experimental pattern which displays the highest " n e t " symmetry has a (6ram) symmetry for the Z O L Z and a 3m symmetry for the whole pattern (Fig. 7a). For this pattern, the reflections of the F O L Z are situated in the centroids of the equilateral triangles formed with the net of the Z O L Z reflections. In addition, no pattern with (4mm), 4mm " n e t " symmetry has been observed. These features indicate, in accordance with Table 7 and Fig. 5b in Ref. [1], that the crystal system is trigonal and the lattice is rhombohedral. From Table 2 of Ref. [1] it is also deduced that the experimental pattern of Fig. 7a corresponds to the [0001] zone axis. For the trigonal system with rhombohedral lattice, the identification of the glide planes requires examination of the (1120) Z A P (see Table
315
,:
• • •
.x,"
o e ~
".'.
\
/",X ..::' . "'.. O0 ,:OO00..'" '.::::. •':::/ :~................ .. ............ O00 O0 • O
• • •
Ir-~ ~' ' .~
"'....
O 0 O. Oo.e • ql, •
• e •
.
....'iJ • • g i g
• o
. I , " ' . " ........(m2)l
_,Z~_.~'~'-.!.: ......... ...• -.-o:'.:'~i~ ~.-.i
\ 'gee~ ~ ' . X ' . \..~
...." • e ° e
'v~-,,..X: .v / o • q~'-.,
oOo°o e
e •
,'","',, ..i:::'; .:::./
. . • ....
~
".., . - . . -
.%~,/
O./O • • • 0 " 4 • • • 0",,0
• • •
ooo..
o.o
OIDOO00
'.qoo
." O 0 0 0 0 e 0 0 0 0 0 0 0 ~
~ O 0 0 0 O : O 0 0 0 0
Figure ZAP
a
Extinction Symbol
R(obv)--
--
b
[0001 ] R(obv)-c
Fig. 8. (a, b) Theoretical [0001] ZAPs related to the R(obv)-and R(obv)-c extinction symbols. Forbidden reflections, labelled f, are situated on the m'1, m'2 and m'3 mirrors in the FOLZ for the R(obv)-cextinction symbol.
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A. Redjai'mia, Z P. Morniroli / Ultramicroscopy 53 (1994) 305-317
the F O L Z reflection nets. In fact, since the lattice is described by means of a triple hexagonal obverse cell, the Laue zone which appears on the diffraction pattern is not the first-order Laue zone but the third-order Laue zone (TOLZ). From comparison with Fig. 9e in Ref. [1], it is concluded that the partial extinction symbol is R ( o b v ) . - . Taking into account Table 3.2 of the International Tables for Crystallography [8], the extinction symbol is R(obv)--. This is in agreement with 5 space groups belonging to 5 point groups as indicated in Table 4. To make the distinction between these five possible space groups the point group must be identified from observation of the "ideal" symmetries. The Z O L Z and whole pattern "ideal" symmetries for the [0001] Z A P are respectively (6) and 3 as illustrated in Fig. 7a (see particularly some examples of difference of intensity between several couples of arrowed reflections which prove the absence of "ideal" mirrors if intensity is taken into account). According to Table 8 in Ref. [1], the point group is 3. This allows one to conclude that the space group for the R phase is R3. The extinction symbol R(obv)-- can also be identified and distinguished from the other possible extinction symbol R(obv)-c, by observing the [0001] ZAP. On this Z A P all the reflections of the F O L Z situated on the three mirrors m'l, m'2 and m'3 are forbidden reflections in the case of the extinction symbol R(obv)-c (Fig. 8b) while no forbidden reflections exist in the case of the symbol R(obv)-- (Fig. 8a). We verified on well oriented diffraction patterns that the reflections of the Z O L Z present on these three mirrors (reflections noted nf in Fig. 7a) are not forbidden reflections. The intensity of these reflections does not decrease or vanish when the m' mirrors are accurately set parallel to the electron beam. This fact confirms that the extinction symbol is R(obv)- -.
4. Conclusions
In this paper, three examples are given in order to explain how microdiffraction patterns
can be used to identify or to characterize crystal structures. A routine method, based on observation of features present in microdiffraction patterns, is proposed. It starts with the observation of the " n e t " symmetries, that is the symmetry of the microdiffraction patterns which only takes into account the position of the reflections. Then, it continues with the observation of the Z O L Z / F O L Z shift a n d / o r the Z O L Z / F O L Z periodicity differences. For each of the three examples given, the observation of these features was usually easy to perform and the deduction of the crystal system, the Bravais lattice and the glide planes was always successful. This is mainly due to the fact that the identification of these features does not necessarily require a perfect orientation of the zone axis with respect to the electron beam. Solutions are given to overcome the difficulties which could be encountered with the frequent absence of F O L Z reflections on high-symmetry zone axis patterns. The derivation of the Buerger extinction symbol results from observation of one, two or at most three different specific zone axis microdiffraction patterns. Obtaining these required zone axis patterns is greatly facilitated by observing, when possible, Kikuchi lines. For some crystal systems, the method requires obervation of zone axes separated by a 90 ° angle. A tilt-rotation or a double tilt specimen holder with large tilt capabilities is therefore recommended. Sometimes it may be necessary to observe a few crystals with various original orientations in order to reach the required zone axes. With single crystals it is necessary to prepare thin foils with definite orientations. Other difficulties may be encountered with textured specimens or with specimens composed of particles that have needle or platelet shapes which adopt a preferential orientation when deposited on a grid. These experimental difficulties can be overcome by obtaining the extinction symbol from observation of a unique zone axis pattern by means of the identification of the forbidden reflections present in the Z O L Z but also in the H O L Z . Nevertheless, since this technique requires a more accurate orientation of the crystal with respect to the
A. Redjaimia, J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317
electron beam, it is more delicate to perform but it could give very efficient and elegant results. The method works with various types of specimen. Of course there is no difficulty with large crystals. With small particles (see the given examples of the o-, X and R phases) or with faulted crystals, the choice of the spot size is crucial and must be correctly adapted to the observed specimen.
This first part allows one to deduce a few possible space groups. Very often the knowledge of these results is sufficient and constitutes a very sure method of identification of a crystal and the analysis can be stopped there. The second part is more delicate since it requires observation of the "ideal" symmetries. Now the intensity of the reflections is taken into account. For that reason a specimen free of defects and also a perfect orientation of the beam with respect to the zone axis, or at least with respect to an "ideal" mirror, is required. The three examples given illustrate how this method can be used to identify the point group and therefore to remove ambiguities between the possible space groups. The example of the cr phase shows the usefulness of combining the results obtained firstly from observation of the " n e t " symmetry with results deduced from C B E D patterns. In that case, which can be used with specimens giving good C B E D patterns, the simplest C B E D experiments can be planned. Usually C B E D patterns containing only 2D information can be used and this constitutes a very important experimental simplification. The method we give here, is a routine method but every determination inevitably constitutes a particular case where all useful information must be taken into acount. Two approaches are generally encountered when dealing with phase identification. The first approach is used when the main aim is to identify a phase among several possible phases whose crystallographic features are well known. Then specific experiments can be performed to verify that the " n e t " and "ideal" symmetries, Z O L F / F O L Z shifts and periodicity differences or BF and WP symmetries of C B E D
317
patterns are in agreement with the possible phases. The second approach, which we have tried to use in this paper, consists of reaching, step by step, a few possible space groups and then removing ambiguities between these possible space groups by indentifying the point group. This approach, if it can be carried out successfully, allows one to determine the crystal structure and also to identify a phase, with certainty. Another important point to mention is that these methods are performed at a microscopic scale. By using modern microscopes which permit the use of very small spot size, they can be applied to very small particles or to faulted crystals. The complementary natures of microdiffraction and C B E D are also pointed out.
5. Acknowledgement The authors would like to thank J.W. Steeds for helpful discussions on microdiffraction.
6. References [1] J.P. Morniroli and J.W. Steeds, Ultramicroscopy 45 (1992) 219. [2] J.W Steeds and N.S. Evans, in: Proc. 38th Annual EMSA Meeting, San Francisco, 1980, Ed. G.W. Bailey, p. 188. [3] M. Raghavan, J. Yu Koo and R. Petkovic-Luton, J. Met. 35 (1984) 6. [4] J.W. Steeds and R. Vincent, J. Appl. Cryst. 16 (1983) 317. [5] M. Raghavan, J.C. Scanlon and J.W. Steeds, Met. Trans. 15A (1984) 1299. [6] Raghavan Ayer, C.F. Klein and L. Angers, in: Proc. l l t h Int. Congr. on Electron Microscopy, Kyoto, 1986, p. 701. [7] N.J. Buerger, Z. Kristallogr. 91 (1935) 255. [8] International Tables for Crystallography, Ed. T. Hahn (Reidel, Dordrecht, 1988). [9] B.F. Buxton, J.A. Eades, J.W. Steeds and G.M. Rackham, Proc. Roy. Soc. (London) A 281 (1976) 171. [10] M. Tanaka, R. Saito and H. Sekii, Acta Cryst. A 39 (1983) 357. [11] A. Redja'imia, Thesis, INPL Nancy, 1991. [12] J.W. Steeds and R. Vincent, J. Microsc. Spectrosc. Electron. 8 (1983) 419. [13] J.W. Steeds, J.R. Baker and R. Vincent, Electron Microscopy 1982, Vol. 1 (1982) 617. [14] J. Gjcnnes and A.F. Moodie, Acta Cryst. 19 (1965) 65.
Ultramicroscopy53 (1994) 305-317
Application of microdiffraction to crystal structure identification A. Redja'imia a, j.p. Morniroli b a Laboratoire de Science et G~nie des Surfaces, URA CNRS 1402, Ecole des Mines, Parc de Saurupt, F-54042 Nancy Cedex, France Laboratoire de M~tallurgie Physique, URA CNRS 234, Universit~ de Lille I, Bdtiment C6, F-59655 Villeneuue d'Ascq Cedex, France
(Received 25 January 1993; in final form 28 December 1993)
Abstract
In a previous paper by J.P. Morniroli and J.W. Steeds [Ultramicroscopy 45 (1992) 219] it was shown that microdiffraction patterns obtained with a nearly parallel electron beam and a small spot size can be used to identify and to characterize crystal structures. A systematic method was proposed to deduce, via the Buerger extinction symbol, the space group or a set of consistent space groups. In order to illustrate how this method can be applied experimentally, three examples concerning the ~, g and R intermetallic phases are detailed. Experimental solutions are given to overcome difficulties connected with the frequent absence of high-order Laue zone reflections on high-symmetry zone axis patterns. The possibility to identify the Buerger extinction symbol from a unique zone axis pattern is also explored.
1. Introduction
In a previous paper by J.P. Morniroli and J.W. Steeds [1] it was shown that microdiffraction patterns obtained with a focused and nearly parallel electron beam allows one to obtain, in a systematic way, the crystal system, the Bravais lattice and to reveal the presence of glide planes. These crystallographic features are easily deduced from observations of three pieces of information present on the microdiffraction patterns, namely: • The " n e t " symmetry, that is the 2D plane symmetry which is concerned with the position of the reflections on the microdiffraction pattern. For each diffraction pattern, the " n e t " symmetry of the zero-order Laue zone ( Z O L Z ) and the " n e t " symmetry of the whole pattern (WP) can be inferred. Both symmetries are in connection with the crystal system.
• The shift between the Z O L Z and the first-order Laue zone (FOLZ) reflection nets. It is in connection with the Bravais lattice [2,3]. • The periodicity difference between the Z O L Z and the F O L Z reflection nets. It is in connection with the presence of glide planes [4-6]. A systematic method is proposed which involves three steps. The deduction of the crystal system constitutes the first step. It is made by searching the experimental microdiffraction pattern which exhibits the highest " n e t " symmetries. The crystal system is then deduced by means of Table 7 in Ref. [1] and the corresponding zone axis by means of Table 2 in Ref. [1]. In a second step, a simultaneous determination of the Bravais lattice and glide planes is made by comparing microdiffraction patterns with theoretical ones. For each crystal system, one,
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A. Redja~mia, J.P. Mornirofi / Ultramicroscopy 53 (1994) 305-317
two or at most three different experimental patterns corresponding to high crystal symmetry elements are required for this analysis, as indicated in Table 5 in Ref. [1]. All these patterns, except for those connected with the monoclinic system, contain at least two orthogonal " n e t " mirrors m, and m 2. In order to facilitate the comparison with the theoretical patterns two features are taken into account: • The smallest rectangles or squares, centered or not, and having their sides parallel to the " n e t " mirrors m I and m 2 are drawn in the Z O L Z and in the F O L Z . They give information about the Z O L Z / F O L Z periodicity difference connected with the presence of glide planes. • The presence or the absence of F O L Z reflections situated on these mirrors m~ and m 2. They give information about the Z O L Z / F O L Z shift connected with the Bravais lattice. • Eventually screw axes can also be identified by specific methods described in Ref. [1]. Comparison with the theoretical patterns leads to the extinction symbol introduced by Buerger [7], which by means of Table 3.2 of the International Tables for Crystallography [8] allows one to deduce a few possibilities of space groups. These experiments are usually easy to perform since they do not require a very accurate orientation of the specimen with respect to the electron beam. They can be carried out successfully with most specimens. The third and last step consists of removing ambiguity between the possible space groups. Depending on the specimen, two ways are possible. If the specimen gives good quality C B E D (convergent b e a m electron diffraction) patterns displaying 3D or at least 2D information, then, these patterns can be used to determine the point groups by means of the methods given by Buxton et al. [9] or by Tanaka et al. [10]. It is pointed out, that the results previously obtained by microdiffraction can be used, in association with Tables 2 - 4 in Ref. [9] or Fig. 3 in Ref. [10], to plan the simplest C B E D experiments to perform. Very often, C B E D patterns only displaying 2D information are sufficient to realize this analysis and this constitutes a very important experimental simplification.
If good C B E D patterns are not available, then an alternative solution consists of examining the "ideal" symmetries of the microdiffraction patterns. The "ideal" symmetry is the 2D plane symmetry which takes into account both the position and the intensity of the reflections on a microdiffraction pattern. As indicated in Table 8 in Ref. [1], this "ideal" symmetry is connected with the point groups and, from observation of specific zone axes, it can be used to identify the point group. Nevertheless, the corresponding experiments are sometimes difficult to perform since they require a perfect orientation of the crystal with respect to the electron beam. In this paper it is intended to show that this approach can be used as a routine technique to identify and characterize phases. To that aim, the examples of the or, X and R intermetallic phases are given. All the notations and the conventions given in Ref. [1] will be used in this paper. Particularly, the "ideal" symmetries and mirrors will be underlined in order to make the distinction with the " n e t " ones and all features related with the Z O L Z will be written between parentheses. For patterns with 2mm " n e t " symmetry containing two perpendicular mirrors, the mirror parallel to the larger side of the rectangle drawn in the F O L Z is named m I and the other m 2. For patterns with 4mm " n e t " symmetry containing two sets of perpendicular mirrors, the two perpendicular mirrors parallel to the smallest square drawn in the F O L Z are named ml, m2 and the other set m' 1, m 2. For patterns with 6mm " n e t " symmetry, the set of three mirrors parallel to the shortest reflections present in the Z O L Z is named m,, m2, m 3 and the second set, rotated at 30 ° with ' m ' 3respect to the first one, is named m'~, m2, Trigonal crystals are described by means of a triple obverse hexagonal cell. !
2. Experimental methods The tr, X and R intermetallic phases observed in this study are present in a duplex austeniticferritic stainless steel as small particles with an average size of about 0.1 /xm [11]. Thin foils for
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A. Redjafmia, J.P. Mornirofi / Ultramicroscopy 53 (1994) 305-317
Incident beam [uvw] Zone Axis Specimen
29~. FOLZ diffracted beams rransm~ed
O*
Tiltajn lie
22.
sphere
~La~er +1 er 0 Layer-1 a 2%,.
Incident beam ~i
~
/
FOLZ
~iffl"actedbeams
Layer -1
b
L
c
Fig. 1. Ewald sphere constructions: (a) The incident beam is parallel to a zone axis giving a symmetrical zone axis pattern. (b) The incident beam is slightly tilted from the zone axis in order to improve the visibility of the FOLZ reflections. (c) The incident beam is tilted with a 20syrn angle from the zone axis so that the first negative layer of the reciprocal lattice becomes tangential to the sphere. A "FOLZ" diffraction pattern is obtained.
A. Redjai'mia, J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317
308
S'¢mm,etrles {N = "Neet" : I = "Ideal')
Sigma
a = I0011
(4,~nt) WP
4ram
b = II(YOl
J ~
e=lllOI N ] (2rnnl) ~ ) 2ran___.! 2ram 2mn.....~l
(2~,n)
I
i 2111111
Extinction Symbol
P-..
P. n
P_-
"~t°
-•Ol ij.....m2
a
[oou ZAP
m'oo,
~
[I 10] Z A P
002 :
. ~il,.m2: t)
{ lOOl Z A P
m: 110
A. Redjai'mia, J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317
electron microscopy are prepared by twin jet electrothinning in a solution of 5% perchloric acid in 95% II-butoxyethanol at 40 V potential. They are observed with the Philips CM12 and CM30 electron microscopes. The small convergence angle is obtained by using a 10 /zm C 2 condensor aperture and the spot size is in the range 10-100 nm. The zone axis patterns are accurately oriented with respect to the electron b e a m by using the dark field deflectors. Frequently, high-symmetry zone axis patterns exhibit large F O L Z rings. As a result, the F O L Z reflections are very weak due to their large 0sy m diffraction angle as shown on the Ewald sphere construction of Fig. la or absent due to the limited acceptance angle of the microscope. Therefore, the " n e t " and "ideal" whole pattern symmetries, as well as the Z O L Z / F O L Z shifts and periodicity differences are not available on these symmetrical patterns. To solve this problem, the operating voltage of the microscope can be reduced so as to decrease the F O L Z radius or the specimen can be cooled to liquid-nitrogen t e m p e r a t u r e to enhance the intensity of the F O L Z reflections. A better solution consists in tilting the specimen slightly around the zone axis until F O L Z areas appear on the pattern (Fig. lb). Due to lower 0 diffraction angles, the F O L Z reflections obtained in that way are usually stronger and more visible. The tilt angle can be chosen so that it equals 20sym. Then, the first negative layer plane of the reciprocal lattice becomes tangential to the Ewald sphere as indicated in Fig. lc and a " F O L Z " microdiffraction pattern is obtained. This experimental solution, which was previously used by Steeds et al. [12,13] to analyze the fine structure of H O L Z reflections, constitutes an elegant
309
Fig. 3. [001] " F O L Z " microdiffraction pattern of o- phase obtained by tilting the incident beam around [001] as indicated in Fig. lc.
method to characterize the Z O L Z / F O L Z and periodicity differences.
shifts
3. Experimental results 3.1. Example I: o" phase The major Z A P s investigated for this phase show that the highest " n e t " symmetry encountered is (4mm), 4mm (Fig. 2a). This pattern was obtained at 120 kV from a specimen cooled at liquid-nitrogen temperature. The F O L Z reflections are clearly observed. At 200 or 300 kV these F O L Z reflections are not visible on a symmetrical pattern but they can be observed by tilting. A " F O L Z " microdiffraction pattern obtained by the method described in Section 2, is given in Fig. 3.
Fig. 2. Microdiffraction patterns for the tetragonal o- phase: (a) [001] zone axis microdiffraction pattern showing (4mm), 4 m m " n e t " and (4mm), 4 m m "ideal" symmetries. The absence of Z O L Z / F O L Z periodicity difference indicates that there is no glide plane perpendicular to the [001] direction and that the partial extinction symbol is P - . . . (b) [100] zone axis microdiffraction pattern showing (2mm), 2mm " n e t " and (2mm), 2mm "ideal" symmetries. T h e periodicity difference between the Z O L Z and the F O L Z reflection nets reveals the presence of a diagonal n glide plane perpendicular to [100]. The deduced partial extinction symbol is P • n •. (c) [110] zone axis microdiffraction pattern showing (2mm), 2mm " n e t " and (2mm), 2mm "ideal" symmetries.The absence of Z O L Z / F O L Z periodicity difference indicates that there is no glide plane perpendicular to the [110] direction and that the partial extinction symbol is P . . - .
A. Redja'[rnia, J.P. Morniroli / Ultrarnicroscopy 53 (1994) 305-317
310
Since no pattern with (6mm) Z O L Z " n e t " symmetry was observed, the crystal system is, according to Tables 7 and 2 in Ref. [1], tetragonal and the zone axis with (4mm), 4mm " n e t " symmetry is [001]. For this crystal system, the ZAPs needed to identify the Bravais lattice and the glide planes are [100], <100) and <110) (Table 5 in Ref. [1]). To compare these experimental patterns with the theoretical ones of Fig. 9c in Ref. [1] the following operations are performed: • The " n e t " mirrors are identified and the F O L Z reflections present on them are distinguished by means of open circles.
• The smallest squares or rectangles (centered or not) whose sides are parallel to these mirrors are drawn both in the F O L Z and in the Z O L Z . For the F O L Z , this operation is very easy to perform on the " F O L Z " microdiffraction pattern of Fig. 3. The [001] pattern (Fig. 2a) contains two sets of perpendicular " n e t " mirrors ml, m 2 and m'1, m'2. F O L Z reflections are situated along each of these mirrors. The smallest squares whose sides are parallel to m], m2 are identical in the F O L Z and in the Z O L Z . The (100) pattern (Fig. 2b) exhibits F O L Z reflections along the two " n e t " mirrors m 1 and
/~i . . . . . . . . . .
":::" ,;:::::~:::::, ~::: ,. : ::": " : :.::::": ; ' " . . . Jmj"::::" . . . " : : : : ":: " : : "~'\ ::: :::: .::::~:. :~: ~:\ :: : : : :::::~:::: ::: :" |oe
•
• ee*
i : : " :::"
• • * * * *i*e~:e**eo
::::":::::
m~ • -**a
,e,l *e~
m2.
"::: "::
\'i!.'~!!:. ":::':::" .:~W.ii ~]
[oo~] zAP
P-n- or P-b-
a
b
Fig. 4. I d e n t i f i c a t i o n of t h e f o r b i d d e n reflections by tilting c a r r e t u l l y t h e s p e c i m e n a l o n g t h e m i r r o r s m I a n d m 2 so as to o b s e r v e
parts of the HOLZs. The TO01 and ]~ 0 2 forbidden reflections are identified in the FOLZ and in the SOLZ along the mirror m 2.
A. Redja?mia, J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317 Table 1 Space groups and point groups for extinction symbol P-nExtinction symbol P-n-
Space groups
Point groups
P42nm P4n2 42 21 2 P. . . m n m
4mm 4m2 4 mm m
.
m 2. The centered rectangle drawn in the ZOLZ has double size as compared with the noncentered rectangle drawn in the FOLZ.
WP Symmetry ZAP I
311
The (110) pattern (Fig. 2c) only displays FOLZ reflections along the m 2 "net" mirror and the same rectangles, with sides parallel to m I and m2, are present in the ZOLZ and in the FOLZ. Comparisons of these three experimental ZAPs (Figs. 2a-2c) with Fig. 9c in Ref. [1] give the following individual extinction symbols: P--for [001], P- nfor (100), P'.for (110). The addition of these three results gives the partial extinction symbol P-n- which according to
2mm [1101 ]
Fig. 5. [110] C B E D pattern displaying 2D information inside the transmitted and diffracted discs and exhibiting a 2mm WP symmetry.
312
A. Redja't'mia, ZP. Morniroli/ Ultramicroscopy 53 (1994) 305-317
T a b l e 3.2 in Ref. [8] is in a g r e e m e n t with 3 space g r o u p s b e l o n g i n g to 3 p o i n t g r o u p s as i n d i c a t e d in T a b l e 1. It is also possible to o b t a i n the P-n- extinction symbol directly on t h e [001] Z A P . T h e t h e o r e t i c a l p a t t e r n which c o r r e s p o n d s to the extinction symbol P-n- (or P-b-) given in Fig. 4a displays forbidd e n reflections in t h e Z O L Z which can be identified by m e a n s of G j 0 n n e s a n d M o o d i e lines [14]. It also c o n t a i n s Okl a n d hOl f o r b i d d e n reflections l o c a t e d in t h e F O L Z a n d in the S O L Z (secondo r d e r L a u e z o n e ) along the " n e t " m i r r o r s m I a n d
m2. T h e s e f o r b i d d e n reflections c a n b e o b s e r v e d on the p e r f e c t l y o r i e n t e d [001] zone axis p a t t e r n . A m u c h b e t t e r solution consists of tilting the s p e c i m e n along the m i r r o r s by m e a n s o f the d a r k - f i e l d b e a m d e f l e c t o r s as shown in Figs. 4b a n d 4c. T h e F O L Z reflections a r e t h e n m o r e visible a n d t h e f o r b i d d e n r e f l e c t i o n s i d e n t i f i e d m o r e surely. T h e distinction b e t w e e n the two extinction symbols P-n- a n d P-b- can b e m a d e by indexing the Okl a n d hOl f o r b i d d e n reflections. In the first case, t h e f o r b i d d e n reflections a r e of t h e type
Symmemcs (N = "Net" ; I = "ldeaI") m'3 Khi ZOLZ WP Extinction Symbol
a = 10011 N [ I
b= Illll N
(4mm)[ (4_..~I1) (6ram) 4ram I 2ram
3m
c = 10111 N (2rean) 2ram
1-.. o r F-..
I..•
m2
,,.,
10 OmZAP
0 iii
;i ii ii~
2OO
a
I0011ZAP
\P
Fig. 6. Microdiffraction patterns of the cubic X phase: (a) [001] zone axis microdiffraction pattern showing (4mm), 4mm "net" and (4mm), 2mm "ideal" symmetries. The absence of ZOLZ/FOLZ periodicity difference indicates that there is no glide plane perpendicular to the [001] direction and that the partial extinction symbol is I-.. or F-... (b) [Ell] zone axis microdiffraction pattern showing (6mm), 3m "net" symmetries. (c) [011] zone axis microdiffraction pattern showing (2mm), 2mm "net" symmetries. The absence of ZOLZ/FOLZ periodicity difference indicates that there is no glide plane perpendicular to the [011] direction and that the partial extinction symbol is I . . - .
A. Redja'imia, J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317
(k + l) 4: 2n or (h + l) 4~ 2n while in the second case they are of the type k 4: 2n or h 4: 2n. The forbidden reflections identified in Fig. 4b have 10 0 1 and 19 0 2 indices. In addition, no forbidden reflections are observed along the mirrors m'1 and m'2 (Fig. 4c). These results are in perfect agreement with the P-n- extinction symbol. The distinction between the three possible space groups requires the identification of the point group. This identification can be inferred from observation of the "ideal" symmetries, that is the symmetries which take into account both the position and the intensity of the reflections on the microdiffraction pattern. The "ideal" symmetries for the [001] Z A P are (4mm), 4mm (Fig. 2a). According to Table 8 in Ref. [1], the possible point groups are then 4mm or 4 / m m m . To remove the ambiguity between these two point groups, observation of the "ideal" symmetry of the (100) or (110) ZAPs is required. The two experimental [100] (Fig. 2b) and [110] (Fig. 2c) ZAPs exhibit (2mm), 2mm "ideal" symmetries. Therefore it is concluded that the point and space groups for the or phase are respectively 4 / m m m and P 4 2 / m 2 1 / n 2 / m . This phase provides an example where microdiffraction can also be used in connection with C B E D to identify the point group. For this phase it was possible to obtain (110) C B E D patterns which display 2D information inside the transmitted and diffracted discs (Fig. 5) and exhibit a 2mm whole pattern symmetry. Since only 2D information is present, the projection diffraction groups must be taken into account. From Tables 3 and 4 in Ref. [9] we can deduce the projection diffraction groups and then the corresponding whole pattern symmetry for the 3 possible point groups as given in the Table 2. The 2mm WP symmetry observed on the experimental pattern of Fig. 5 is in agreement with the 4 / m m m point group. Therefore the space group for the ~ phase is P 4 z / m 2 t / n 2 / m . It is pointed out that a C B E D pattern which only exhibits 2D features inside the discs allows one, in connection with microdiffraction patterns, to identify uniquely the point and space group of this phase. Since C B E D with 2D features are easier to obtain than C B E D patterns with 3D
313
Table 2 Projection diffraction group and whole pattern symmetry Point group Projection WP symmetry Diffraction group 4mm m1R m 7,2m ml R m 4 -- mm 2mm1R 2mm m
information, this constitutes an important experimental simplification.
3.2. Example II." X phase The highest " n e t " symmetries observed for this phase are (4mm), 4mm (Fig. 6a) and (6mm), 3m (Fig. 6b) which, according to Table 7 in Ref. [1], correspond to a cubic system. One notices that the identification of the cubic system does not necessarily require the observation of the 4mm and 3m whole pattern " n e t " symmetries. This system can be identified, in a simplified way, only from the (4mm) and (6mm) Z O L Z " n e t " symmetries. The zone axes with (4mm), 4mm and (6mm), 3m symmetries respectively correspond to the (001) and (111) Z A P s as indicated in Table 2 in Ref. [1]. The specific Z A P s to observe for Bravais lattice and glide plane identifications are (001) (Fig. 2a) and (110) (Fig. 2c) (Table 5 in Ref. [1]). On the (001) ZAP, two sets of perpendicular " n e t " mirrors ml, m 2 and m'~, m'2 are identified. The smallest squares drawn in the Z O L Z and in the F O L Z have their sides parallel to the m~, m 2 mirrors and are identical. F O L Z reflections are present on the m'~, m'2 mirrors but absent on ml, m 2 mirrors. In agreement with Fig. 9d in Ref. [1], the corresponding individual partial extinction symbols is either I -.o or F - - . . Therefore, the distinction between the I and F Bravais lattices is not possible from observation of this unique (001) ZAP. The (110) Z A P (Fig. 6c) exhibits (2mm), 2mm " n e t " symmetries. No F O L Z reflections are present on either of the two perpendicular " n e t " m 1 and m 2 mirrors and the two rectangles with sides parallel to the mirrors drawn in the Z O L Z and in
314
A. Redja'fmia, J.P. Morn iroli/Ultramicroscopy 53 (1994) 305-317
the F O L Z are identical. Comparison with Fig. 9d in Ref. [1] leads to the individual partial extinction symbol I • .-. Contrary to the (001) ZAP, the body centered Bravais lattice is uniquely identified from this (110) ZAP. Addition of the two individual partial extinction symbols leads to I--- which is in agreement with the 7 space groups associated with the 5 point groups (Table 3.2 in Ref. [8]) given in Table 3. To make the distinction between these seven
possible space groups, the point group must be identified. This can be realized from observation of the (001) Z A P "ideal" symmetry as indicated in Table 8 in Ref. [1]. This (001) pattern (Fig. 6a) exhibits a (4mm) "ideal" Z O L Z symmetry. However, careful examination of the intensity of certain couples of reflections (see particularly the couples of reflections arrowed on Fig. 6a) reveals that the WP "ideal" symmetry is only 2mm since the two " n e t " mirrors m t and m 2 are not "ideal" mirrors if the intensity of the reflections is taken
OlT2~ OTll
13 [iltOlZAP
Synlnlctllex (N = "Net" : I = "ldeal"~
R phase
IT 2TOl
m2
........ : 1210 a
ZOLZ WP Extinction Symbol
a = 100011 (6n~mt
3m
I (6) ~
b =121101 N t21 2 RIob~ J
m]
[0001 ] Z A P
Fig. 7. Microdiffraction patterns of the trigona] R phase: (a) [0001] zone axis microdiffraction pattern showing (6ram) and 3m " n e t " and (6), _3 "ideal" symmetries. (b) [2110] zone axis microdiffraction pattern, showing (2) and 2 " n e t " symmetries. It reveals no periodicity difference between thc Z O L Z and the FOLZ. The deduced extinction symbol is R(obv)--.
A. Redjai'mia,J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317 Table 3 Space groups and point groups for extinction symbol I--Extinction Space groups Point groups 123 23 1213
I---
Im3
m3
I432 14132
432
I43m
43m
4_2 I--3-m m
4_2 --3-m m
5 in Ref. [1]). Analysis of the [ff.ll0] ZAP, which exhibits (2), 2 " n e t " symmetries (Fig. 7b), reveals no periodicity difference between the Z O L Z and
::::,:?:': :':':;*:!:: "" A:;:".. ' " " ::;:X • 0 • •
• •
"410 • O 0 0 O 0 0 0 l t / • 0"41 • O 0 O,O O 0 • (~"O • O
° o' e. •
oO00 • •
• OOII • • e
"i, • o!o • ; "......
......
• •
O0000
0 0 O
• 0
into account. According to Table 8 in Ref. [1], the point group corresponding to (4mm), 2mm "ideal" symmetries is 43m. It is concluded that the space group of this investigated X phase is I43m.
.."0
O O •
o o
.,"
• '
• • • co°ego
OOq)0 • •
.......ii!i!1
• O0000 • • •
O •
oo
O •
•
0
..
,."'el 0 0
\'.'..
0 O O O O • q)"..~
O O B 0 O0
.
•
:
-.. ,!._._._.'
0 @ • .," • 410,""
~.-~
¢-. .
01111
eo
/
/
!
• • •
'~..
o-~..o e
,_ _
~
\
\
/ ".'_.'..~G\ ."
.
".'A
~Oe
.,.v_.-_._ ".',~,X
--
i
•
I110
. - "-".v.\ "'o',
e e ~ o e e e o ' e o e e t
I ";'~.'.'." ', • e
~
'~
...':
O"O..
,%'7
, 0 • 0 " • • • 0 , - , - , ~
/::" ',, . . . .f. .. 0".-" 't,o,,'n,,,,¥
/
•
41 41
O0 • . "~ 0 •
m~' " Z ~
,~-.,.
o •
• •
00000~0090
/
O •
".
• 0
.
•
410
0 • O"...
• •.
0 O0 . 0 O
O
• oo
"\
,,,,: - - , 0 0 O0
O0
' "'"' O
~
."
3.3. Example III: R phase
Table 4 Space groups and point groups for extinction symbol R(obv)-Extinction symbol Space groups Point groups R3 3 R3 R(obv)-R32 32 R3m 3m R3m 3m
°..~i~°
ql0 • • • OoO004 ,, .-" oOoeoOeOoOo • • ' i • • •
0 • • •
.. j..:::::...............
O •
For the R phase, the experimental pattern which displays the highest " n e t " symmetry has a (6ram) symmetry for the Z O L Z and a 3m symmetry for the whole pattern (Fig. 7a). For this pattern, the reflections of the F O L Z are situated in the centroids of the equilateral triangles formed with the net of the Z O L Z reflections. In addition, no pattern with (4mm), 4mm " n e t " symmetry has been observed. These features indicate, in accordance with Table 7 and Fig. 5b in Ref. [1], that the crystal system is trigonal and the lattice is rhombohedral. From Table 2 of Ref. [1] it is also deduced that the experimental pattern of Fig. 7a corresponds to the [0001] zone axis. For the trigonal system with rhombohedral lattice, the identification of the glide planes requires examination of the (1120) Z A P (see Table
315
,:
• • •
.x,"
o e ~
".'.
\
/",X ..::' . "'.. O0 ,:OO00..'" '.::::. •':::/ :~................ .. ............ O00 O0 • O
• • •
Ir-~ ~' ' .~
"'....
O 0 O. Oo.e • ql, •
• e •
.
....'iJ • • g i g
• o
. I , " ' . " ........(m2)l
_,Z~_.~'~'-.!.: ......... ...• -.-o:'.:'~i~ ~.-.i
\ 'gee~ ~ ' . X ' . \..~
...." • e ° e
'v~-,,..X: .v / o • q~'-.,
oOo°o e
e •
,'","',, ..i:::'; .:::./
. . • ....
~
".., . - . . -
.%~,/
O./O • • • 0 " 4 • • • 0",,0
• • •
ooo..
o.o
OIDOO00
'.qoo
." O 0 0 0 0 e 0 0 0 0 0 0 0 ~
~ O 0 0 0 O : O 0 0 0 0
Figure ZAP
a
Extinction Symbol
R(obv)--
--
b
[0001 ] R(obv)-c
Fig. 8. (a, b) Theoretical [0001] ZAPs related to the R(obv)-and R(obv)-c extinction symbols. Forbidden reflections, labelled f, are situated on the m'1, m'2 and m'3 mirrors in the FOLZ for the R(obv)-cextinction symbol.
316
A. Redjai'mia, Z P. Morniroli / Ultramicroscopy 53 (1994) 305-317
the F O L Z reflection nets. In fact, since the lattice is described by means of a triple hexagonal obverse cell, the Laue zone which appears on the diffraction pattern is not the first-order Laue zone but the third-order Laue zone (TOLZ). From comparison with Fig. 9e in Ref. [1], it is concluded that the partial extinction symbol is R ( o b v ) . - . Taking into account Table 3.2 of the International Tables for Crystallography [8], the extinction symbol is R(obv)--. This is in agreement with 5 space groups belonging to 5 point groups as indicated in Table 4. To make the distinction between these five possible space groups the point group must be identified from observation of the "ideal" symmetries. The Z O L Z and whole pattern "ideal" symmetries for the [0001] Z A P are respectively (6) and 3 as illustrated in Fig. 7a (see particularly some examples of difference of intensity between several couples of arrowed reflections which prove the absence of "ideal" mirrors if intensity is taken into account). According to Table 8 in Ref. [1], the point group is 3. This allows one to conclude that the space group for the R phase is R3. The extinction symbol R(obv)-- can also be identified and distinguished from the other possible extinction symbol R(obv)-c, by observing the [0001] ZAP. On this Z A P all the reflections of the F O L Z situated on the three mirrors m'l, m'2 and m'3 are forbidden reflections in the case of the extinction symbol R(obv)-c (Fig. 8b) while no forbidden reflections exist in the case of the symbol R(obv)-- (Fig. 8a). We verified on well oriented diffraction patterns that the reflections of the Z O L Z present on these three mirrors (reflections noted nf in Fig. 7a) are not forbidden reflections. The intensity of these reflections does not decrease or vanish when the m' mirrors are accurately set parallel to the electron beam. This fact confirms that the extinction symbol is R(obv)- -.
4. Conclusions
In this paper, three examples are given in order to explain how microdiffraction patterns
can be used to identify or to characterize crystal structures. A routine method, based on observation of features present in microdiffraction patterns, is proposed. It starts with the observation of the " n e t " symmetries, that is the symmetry of the microdiffraction patterns which only takes into account the position of the reflections. Then, it continues with the observation of the Z O L Z / F O L Z shift a n d / o r the Z O L Z / F O L Z periodicity differences. For each of the three examples given, the observation of these features was usually easy to perform and the deduction of the crystal system, the Bravais lattice and the glide planes was always successful. This is mainly due to the fact that the identification of these features does not necessarily require a perfect orientation of the zone axis with respect to the electron beam. Solutions are given to overcome the difficulties which could be encountered with the frequent absence of F O L Z reflections on high-symmetry zone axis patterns. The derivation of the Buerger extinction symbol results from observation of one, two or at most three different specific zone axis microdiffraction patterns. Obtaining these required zone axis patterns is greatly facilitated by observing, when possible, Kikuchi lines. For some crystal systems, the method requires obervation of zone axes separated by a 90 ° angle. A tilt-rotation or a double tilt specimen holder with large tilt capabilities is therefore recommended. Sometimes it may be necessary to observe a few crystals with various original orientations in order to reach the required zone axes. With single crystals it is necessary to prepare thin foils with definite orientations. Other difficulties may be encountered with textured specimens or with specimens composed of particles that have needle or platelet shapes which adopt a preferential orientation when deposited on a grid. These experimental difficulties can be overcome by obtaining the extinction symbol from observation of a unique zone axis pattern by means of the identification of the forbidden reflections present in the Z O L Z but also in the H O L Z . Nevertheless, since this technique requires a more accurate orientation of the crystal with respect to the
A. Redjaimia, J.P. Morniroli / Ultramicroscopy 53 (1994) 305-317
electron beam, it is more delicate to perform but it could give very efficient and elegant results. The method works with various types of specimen. Of course there is no difficulty with large crystals. With small particles (see the given examples of the o-, X and R phases) or with faulted crystals, the choice of the spot size is crucial and must be correctly adapted to the observed specimen.
This first part allows one to deduce a few possible space groups. Very often the knowledge of these results is sufficient and constitutes a very sure method of identification of a crystal and the analysis can be stopped there. The second part is more delicate since it requires observation of the "ideal" symmetries. Now the intensity of the reflections is taken into account. For that reason a specimen free of defects and also a perfect orientation of the beam with respect to the zone axis, or at least with respect to an "ideal" mirror, is required. The three examples given illustrate how this method can be used to identify the point group and therefore to remove ambiguities between the possible space groups. The example of the cr phase shows the usefulness of combining the results obtained firstly from observation of the " n e t " symmetry with results deduced from C B E D patterns. In that case, which can be used with specimens giving good C B E D patterns, the simplest C B E D experiments can be planned. Usually C B E D patterns containing only 2D information can be used and this constitutes a very important experimental simplification. The method we give here, is a routine method but every determination inevitably constitutes a particular case where all useful information must be taken into acount. Two approaches are generally encountered when dealing with phase identification. The first approach is used when the main aim is to identify a phase among several possible phases whose crystallographic features are well known. Then specific experiments can be performed to verify that the " n e t " and "ideal" symmetries, Z O L F / F O L Z shifts and periodicity differences or BF and WP symmetries of C B E D
317
patterns are in agreement with the possible phases. The second approach, which we have tried to use in this paper, consists of reaching, step by step, a few possible space groups and then removing ambiguities between these possible space groups by indentifying the point group. This approach, if it can be carried out successfully, allows one to determine the crystal structure and also to identify a phase, with certainty. Another important point to mention is that these methods are performed at a microscopic scale. By using modern microscopes which permit the use of very small spot size, they can be applied to very small particles or to faulted crystals. The complementary natures of microdiffraction and C B E D are also pointed out.
5. Acknowledgement The authors would like to thank J.W. Steeds for helpful discussions on microdiffraction.
6. References [1] J.P. Morniroli and J.W. Steeds, Ultramicroscopy 45 (1992) 219. [2] J.W Steeds and N.S. Evans, in: Proc. 38th Annual EMSA Meeting, San Francisco, 1980, Ed. G.W. Bailey, p. 188. [3] M. Raghavan, J. Yu Koo and R. Petkovic-Luton, J. Met. 35 (1984) 6. [4] J.W. Steeds and R. Vincent, J. Appl. Cryst. 16 (1983) 317. [5] M. Raghavan, J.C. Scanlon and J.W. Steeds, Met. Trans. 15A (1984) 1299. [6] Raghavan Ayer, C.F. Klein and L. Angers, in: Proc. l l t h Int. Congr. on Electron Microscopy, Kyoto, 1986, p. 701. [7] N.J. Buerger, Z. Kristallogr. 91 (1935) 255. [8] International Tables for Crystallography, Ed. T. Hahn (Reidel, Dordrecht, 1988). [9] B.F. Buxton, J.A. Eades, J.W. Steeds and G.M. Rackham, Proc. Roy. Soc. (London) A 281 (1976) 171. [10] M. Tanaka, R. Saito and H. Sekii, Acta Cryst. A 39 (1983) 357. [11] A. Redja'imia, Thesis, INPL Nancy, 1991. [12] J.W. Steeds and R. Vincent, J. Microsc. Spectrosc. Electron. 8 (1983) 419. [13] J.W. Steeds, J.R. Baker and R. Vincent, Electron Microscopy 1982, Vol. 1 (1982) 617. [14] J. Gjcnnes and A.F. Moodie, Acta Cryst. 19 (1965) 65.