Phason-strain identification for quasicrystals by high-resolution electron microscopy

Ultramicroscopy 45 (1992) 299-305 North-Holland

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Phason-strain identification for quasicrystals by high-resolution electron microscopy * F . H . Li, G . Z . P a n , D . X . H u a n g Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China

H. H a s h i m o t o

a n d Y. Y o k o t a

Okayama Unicersity of Science, 1-1 Ridai-cho, Okayama 700, Japan

Received 4 June 1992

The simulation of high-resolution electron microscope images and corresponding electron diffraction patterns for perfect quasicrystals and quasicrystals with phason strain is carried out on the basis of high-dimensional crystal which gives a quasicrystal by cutting it with the physical space. The image-contrast change with the change of defocus and phason-strain strength is shown. The stimulated image for a phason-defected quasicrystal of AI-Cu-Li alloy is in good agreement with the experimental image. It is demonstrated that the local phason strain in quasicrystal can be identified by image simulation, even under the weak phase object approximation.

1. Introduction In m a n y cases, quasicrystal is not perfect, owing to the p h a s o n strain frozen in it [1-3]. Q u a sicrystal distorted by introducing a special linear p h a s o n strain can be treated as an intermediate state between perfect quasicrystal and a crystalline approximant. I n t e r m e d i a t e states between icosahedral quasicrystal and body-centered-cubic (bcc) or almost bcc crystal were found in A 1 - C u Li [4], A I - C u - M g [5,6] and A 1 - M n - S i [7] systems; those between octagonal quasicrystal and cubic crystal were found in the M n - S i system [8]. I n t e r m e d i a t e states were also found between decagonal quasicrystals and their crystalline approximants [9,10]. T h e distorted electron diffraction patterns ( E D P ) of such quasicrystals with a linear phason strain can easily be simulated, because generally the n u m b e r of n o n - z e r o elements

* This project is partially supported by the National Natural Science Foundation of China.

in the phason matrix is limited. In fact, the elements of the phason matrix and hence the phason strain can be d e t e r m i n e d from E D P , provided the crystalline approximant into which the quasicrystal is transforming is known. Evidently, the phason strain d e t e r m i n e d from E D P possesses an average character. High-resolution electron microscope images can show the change of the phason strain from area to area in a quasicrystal. In the present p a p e r a m e t h o d of local phasonstrain identification by high-resolution electron microscope image simulation is reported.

2. Principle T h e r e are two approaches to high-resolution electron microscope image simulation for quasicrystals. O n e is based on the construction of a fictitious crystal with a huge unit cell which consists of a part of the quasicrystal [11,12]. A n o t h e r a p p r o a c h is based on high-dimensional ( H D ) crystal [13] which gives quasicrystal by cutting it

0304-3991/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

F.H. Li et aL / Phason-strainidentificationfor quasicrystalsby HREM

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with the three-dimensional (3D) hyperplane - the physical space (Ell). Evidently, the former approach leads to tedious work, even when the quasicrystal is perfect. For quasicrystals with phason strain it is much more difficult, however not impossible, to construct the corresponding periodic 3D crystal. In the following it will be shown that image simulation for both perfect and phason-defected quasicrystals by means of HD crystal is straightforward. The HD crystal s t r u c t u r e t~(rll , r L) can be written as the convolution of the H D lattice function L(rll, r L) with the function ~h0(ru, rL), which describes the atomic structure inside the H D unit cell: ~b(rll, r L) = S(rL) * L ( r u , r L) * ~b0(rll, r L),

(1) where rll and r L denote the coordinate vectors in physical space and its complementary space pseudo-space (EL), respectively. ~bo(ru, r L) and ~b(ru, r L) are potential distribution functions. The function S(r 1) describes the shape of lattice nodes and the shape of hyperatoms in the pseudo-space. The intersection of any hyperatom with Ell gives the real atom. The expression for the 3D quasicrystal structure P(rll), which is the intersection of ~b(rll, r L) with Eli, can be obtained by firstly carrying out the Fourier transform (FT) of ~b(rll, r L), then projecting the product along E L and finally performing the inverse FT [14].

P(rll ) = ~_, Y'.S(GL)Fo(GII, G L) exp(2rrirliGli), GII G_

(2) where F0(Gll , G L) -- Fo(G) is the FT of the function ~bo(rll, r L) and s(G L) is the FT of S(rL). The quasicrystal structure factor is

F(G) =S(G L)Fo(G ) .

(3)

Hence, there is a one-to-one correspondence between the structure factor of quasicrystal and the structure factor of H D crystal. The structure factor of the quasicrystal equals the structure factor of the H D crystal modulated by the function

s(G L)"

When atoms are of different shape and size in the HD space, the H D crystal structure is described as ~b(rll, r ± ) = Y'~&j(rll, r±) * Sj(r±),

(4)

J

where q~j(rll, r . ) is the potential distribution function of the j-kind of atoms inside the HD unit cell, Sj(r ±) is the window function describing the shape and size of the j-kind of atoms. Let si(G x ) denote the FT o f Si(r x) and fj(Gtl) the atomic scattering factor of the j-kind of atoms. Thus the structure factor of quasicrystals is written as

F(G) = ~,sj(G L) Y'.fj(GII) exp(27rir~nG), j n

(5) with n denoting the order of atoms. The linear phason strain is expressed as w(r±)

= Mrll,

(6)

where M is a second-rank tensor, or M = e iMell,

(7)

with ell and e ± being unit vectors in Eli and E ±, respectively; M is a matrix of order 3 × (n - 3) and is called the phason matrix. When the quasicrystal structure is distorted by a linear phason strain, the structure factor can be derived by a similar procedure as for that of a perfect quasicrystal. The difference is only in the projection direction of the FT of the H D crystal structure. The FT of the H D crystal structure should be projected along the newly defined pseudo-space E'~, but not E ± [14]. The expression of the structure factor of the phason-defected quasicrystal structure is

F'(G) =s(G ±) Y'~fj(GII- G ±IVI) exp(27rirjG), J

(s) when all atoms are of the same shape and size, or is

F'(C,) = •si(G i ) f j ( G i i - G ±M) Eexp(2~irj, G), j n

(9)

F.H. Li et al. / Phason-strain identification for quasicrystals by HREM

m=O.O

301

m=-O.1 Defocus

-20 nm

-100

nm

-120

nm

Fig. 1. S i m u l a t e d i m a g e s o f A 1 - C u - L i icosahedral quasicrystal. T h e incident b e a m is parallel to the fivefold axis. T h e left c o l u m n is for the perfect quasicrystal and the right c o l u m n is for the p h a s o n - d e f e c t e d quasicrystal with rn = - 0 . 1 . D e f o c u s values are - 2 0 (top), - 100 ( m i d d l e ) and - 120 nm. Jags are indicated by arrows.

F.H. Li et al. / Phason-strain identification for quasicrystals by HREM

302

when the atoms are of different shape and size, where IVI denotes the transpose of M. Principally, the H D hyper atoms are in the shape of a polyhedron of which the symmetry is consistent with that of the quasicrystal. For example, in order to obtain the 3D Penrose tiling [15], all HD lattice nodes should be in the shape of a unit triacontahedron. However, for simplicity the H D atoms can be approximated to a sphere or a spherical shell. The image intensity for quasicrystals under the weak phase object approximation is expressed as I = 1 - 2~r~-- 1[ F ( sc, r/)W(sc, r/)],

(10)

where W(~:, rl) denotes the contrast transfer function and 9 - - t is the operator of the inverse Fourier transform. Eq. (10) is the same as that for

a

crystals except that here the parameters ~c and 77 are not integers.

3. Image simulation for AI-Cu-Li quasicrystal It was reported that the icosahedral quasicrystal T2-A16CuLi 3 and the bcc crystal R-AIsCuLi 3 correspond to the same 6D hypercubic crystal with 27 atoms inside the 6D unit cell [16]. One .(A1,Cu) atom is located at the origin, six (A1,Cu) atoms are at edge centers. Twenty-seven Li atoms sit on the 3D hyperplane diagonals. Every two Li atoms divide a 3D hyperplane diagonal into three segments of length portion ( 1 / r 2 ) : ( 1 / " r 3) :(1/'r2), with 7 being the golden mean ( = (1 + x/5-)/2). A good fit between the calculated and observed

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q u a s i c r y s t a l with (a) m = 0, (b) m = - 0 . 1 5 , (c) m = - 0 . 2 0 a n d (d) m = - 0 . 2 3 6 . T h e i n c i d e n t b e a m is p a r a l l e l to t h e fivefold axis.

F.H. Li et al. / Phason-strain identification for quasicrystals by HREM

E D P was obtained by approximating the shape of all atoms as a sphere with a 30% smaller radius than the radius of the sphere with its volume equal to the volume of the unit triacontahedron. This model was used for simulating images of perfect and phason-defected A 1 - C u - L i quasicrystals. The electron-optical parameters are as follows: the accelerating voltage is 400 kV, the

m--0.0

m=-0.20

303

spherical aberration coefficient is 1.0 mm, and the radius of the objective lens aperture is 0.55 .~-1. The phason matrix has the following form: M=

m 0

.

(11)

When m = 0, the quasicrystal is perfect. When

n 15

m---0.236

Fig. 3. Simulated images of A 1 - C u - L i quasicrystal corresponding to EDP's shown in figs. 2a to 2d, respectively. Jags in (b) and (c) are pointed out by arrows and the 1D periodicity in (d) is indicated by bars.

F.H. Li et al. / Phason-strain identification for quasicrystals by HREM

304

m = --0.236 the quasicrystal transforms into the bcc crystal. Fig. 1 shows the simulated images for a perfect quasicrystal projected along the fivefold axis in the left column and for a phason-defected quasicrystal with m = - 0 . 1 in the right column. The defocus values from the top to the bottom are - 20, - 100 and - 120 nm, respectively. Although the image contrast changes evidently with the change of the defocus value, all images in the left column show a perfect fivefold symmetry, while in the right column many jags appear, as pointed out by arrows, which indicates the degeneration of the fivefold symmetry. Such jags are frequently seen in published experimental images. Figs. 2a to 2d are calculated electron diffraction patterns of A 1 - C u - L i quasicrystal with the incident b e a m parallel to the fivefold axis for m = 0, - 0.15, - 0.20 and - 0.236, respectively. The diffraction pattern in fig. 2a shows a perfect fivefold symmetry. The diffraction spots, especially the weak spots, displace along the horizontal gradually with the increase of m from fig. 2a to fig. 2d, and finally the 1D periodicity is formed in fig. 2d. Figs. 3a to 3d are simulated images corresponding to figs. 2a to 2d, respectively. The defocus value is - 40 nm for all images, which is close to the Scherzer focus for the present case. The image in fig. 3a shows a perfect fivefold symmetry. Straight continuous dark bands intersecting one another in an angle of 72 ° can be seen. The distance between adjacent bands obeys the golden ratio. When m = - 0 . 1 5 , the dark bands are broken and jags appear everywhere (fig. 3b). But the vertical dark bands remain continuous and some of them are located equidistantly. This indicates the formation of a local one-dimensional (1D) periodicity. With the further increase of m the local 1D periodicity becomes more clear (fig. 3c). When m = - 0 . 2 3 6 , white dot strings arranged equidistantly along the vertical become more distinct than the vertical dark bands, and a perfect 1D periodicity along the horizontal, as labeled by dark bars, can be seen. This indicates that the quasicrystal is transformed into the bcc crystal. The horizontal direction is parallel to one of the basis vectors of the bcc phase. The image shows i

Fig. 4. (a) Experimental and (b) simulated image of phasondefected A1-Cu-Li quasicrystal. The incident beam is parallel to the fivefoldaxis. m = - 0.10. only the 1D periodicity, because the vertical is not coincident w i t h any zone axis but parallel to the [01r] direction of the bcc phase. Fig. 4a is an experimental image of A 1 - C u - L i quasicrystal taken with the JEM-4000EX electron microscope, and the incident beam is parallel to the fivefold axis. It is obvious that the fivefold symmetry is degenerated. For instance, the figure at the center consists of a small white circle surrounded by ten white dots, and around it there are six such figures, as pointed out by arrows, instead of five. It is interesting that the image fits well to the simulated image with m = - 0 . 1 and an underfocus of 120 nm (fig. 3b).

4. Concluding remarks The high-resolution electron microscope images of a quasicrystal with linear phason strain

F.H. Li et al. / Phason-strain identification for quasicrystals by HREM c a n b e s i m u l a t e d o n t h e basis o f a H D crystal. The image simulation under the WPOA and the a p p r o x i m a t i o n o f H D a t o m s to a s p h e r e gives a g o o d fit to t h e e x p e r i m e n t a l i m a g e . H e n c e , t h e local p h a s o n s t r a i n in q u a s i c r y s t a l s c a n b e i d e n t i f i e d by this m e t h o d . T h e i m a g e s i m u l a t i o n for p h a s o n - d e f e c t e d q u a s i c r y s t a l s c a n also b e c a r r i e d o u t by c o n s i d e r i n g t h e d y n a m i c a l s c a t t e r i n g , as in ref. [13] f o r p e r f e c t q u a s i c r y s t a l s .

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